id sid tid token lemma pos bracis-19045 1 1 a a DET bracis-19045 1 2 kullback kullback NOUN bracis-19045 1 3 - - PUNCT bracis-19045 1 4 leibler leibler NOUN bracis-19045 1 5 divergence divergence NOUN bracis-19045 1 6 - - PUNCT bracis-19045 1 7 based base VERB bracis-19045 1 8 locally locally ADV bracis-19045 1 9 linear linear ADJ bracis-19045 1 10 embedding embed VERB bracis-19045 1 11 method method NOUN bracis-19045 1 12 : : PUNCT bracis-19045 1 13 a a DET bracis-19045 1 14 novel novel ADJ bracis-19045 1 15 parametric parametric ADJ bracis-19045 1 16 approach approach NOUN bracis-19045 1 17 for for ADP bracis-19045 1 18 cluster cluster NOUN bracis-19045 1 19 analysis analysis NOUN bracis-19045 1 20 | | NOUN bracis-19045 1 21 springer springer NOUN bracis-19045 1 22 nature nature NOUN bracis-19045 1 23 link link PROPN bracis-19045 1 24 ( ( PUNCT bracis-19045 1 25 formerly formerly ADV bracis-19045 1 26 springerlink springerlink NOUN bracis-19045 1 27 ) ) PUNCT bracis-19045 1 28 skip skip VERB bracis-19045 1 29 to to ADP bracis-19045 1 30 main main ADJ bracis-19045 1 31 content content NOUN bracis-19045 1 32 advertisement advertisement NOUN bracis-19045 1 33 log log NOUN bracis-19045 1 34 in in ADP bracis-19045 1 35 menu menu NOUN bracis-19045 1 36 find find VERB bracis-19045 1 37 a a DET bracis-19045 1 38 journal journal NOUN bracis-19045 1 39 publish publish VERB bracis-19045 1 40 with with ADP bracis-19045 1 41 us we PRON bracis-19045 1 42 track track VERB bracis-19045 1 43 your your PRON bracis-19045 1 44 research research NOUN bracis-19045 1 45 search search NOUN bracis-19045 1 46 cart cart NOUN bracis-19045 1 47 home home NOUN bracis-19045 1 48 intelligent intelligent ADJ bracis-19045 1 49 systems system NOUN bracis-19045 1 50 conference conference NOUN bracis-19045 1 51 paper paper NOUN bracis-19045 1 52 a a DET bracis-19045 1 53 kullback kullback NOUN bracis-19045 1 54 - - PUNCT bracis-19045 1 55 leibler leibler NOUN bracis-19045 1 56 divergence divergence NOUN bracis-19045 1 57 - - PUNCT bracis-19045 1 58 based base VERB bracis-19045 1 59 locally locally ADV bracis-19045 1 60 linear linear ADJ bracis-19045 1 61 embedding embed VERB bracis-19045 1 62 method method NOUN bracis-19045 1 63 : : PUNCT bracis-19045 1 64 a a DET bracis-19045 1 65 novel novel ADJ bracis-19045 1 66 parametric parametric ADJ bracis-19045 1 67 approach approach NOUN bracis-19045 1 68 for for ADP bracis-19045 1 69 cluster cluster NOUN bracis-19045 1 70 analysis analysis NOUN bracis-19045 1 71 conference conference NOUN bracis-19045 1 72 paper paper NOUN bracis-19045 1 73 first first ADV bracis-19045 1 74 online online ADV bracis-19045 1 75 : : PUNCT bracis-19045 1 76 28 28 NUM bracis-19045 1 77 november november NOUN bracis-19045 1 78 2021 2021 NUM bracis-19045 1 79 pp pp ADP bracis-19045 1 80 406–420 406–420 NUM bracis-19045 1 81 cite cite VERB bracis-19045 1 82 this this DET bracis-19045 1 83 conference conference NOUN bracis-19045 1 84 paper paper NOUN bracis-19045 1 85 access access NOUN bracis-19045 1 86 provided provide VERB bracis-19045 1 87 by by ADP bracis-19045 1 88 university university PROPN bracis-19045 1 89 of of ADP bracis-19045 1 90 notre notre PROPN bracis-19045 1 91 dame dame PROPN bracis-19045 1 92 hesburgh hesburgh PROPN bracis-19045 1 93 library library PROPN bracis-19045 1 94 download download PROPN bracis-19045 1 95 book book NOUN bracis-19045 1 96 pdf pdf PROPN bracis-19045 1 97 download download NOUN bracis-19045 1 98 book book NOUN bracis-19045 1 99 epub epub PROPN bracis-19045 1 100 intelligent intelligent ADJ bracis-19045 1 101 systems system NOUN bracis-19045 1 102 ( ( PUNCT bracis-19045 1 103 bracis bracis NOUN bracis-19045 1 104 2021 2021 NUM bracis-19045 1 105 ) ) PUNCT bracis-19045 1 106 a a DET bracis-19045 1 107 kullback kullback NOUN bracis-19045 1 108 - - PUNCT bracis-19045 1 109 leibler leibler NOUN bracis-19045 1 110 divergence divergence NOUN bracis-19045 1 111 - - PUNCT bracis-19045 1 112 based base VERB bracis-19045 1 113 locally locally ADV bracis-19045 1 114 linear linear ADJ bracis-19045 1 115 embedding embed VERB bracis-19045 1 116 method method NOUN bracis-19045 1 117 : : PUNCT bracis-19045 1 118 a a DET bracis-19045 1 119 novel novel ADJ bracis-19045 1 120 parametric parametric ADJ bracis-19045 1 121 approach approach NOUN bracis-19045 1 122 for for ADP bracis-19045 1 123 cluster cluster NOUN bracis-19045 1 124 analysis analysis NOUN bracis-19045 1 125 download download NOUN bracis-19045 1 126 book book NOUN bracis-19045 1 127 pdf pdf PROPN bracis-19045 1 128 download download NOUN bracis-19045 1 129 book book PROPN bracis-19045 1 130 epub epub PROPN bracis-19045 1 131 alexandre alexandre PROPN bracis-19045 1 132 l. l. PROPN bracis-19045 1 133 m. m. PROPN bracis-19045 1 134 levada levada PROPN bracis-19045 1 135     SPACE bracis-19045 1 136 orcid orcid PROPN bracis-19045 1 137 : : PUNCT bracis-19045 1 138 orcid.org/0000-0001-8253-272910 orcid.org/0000-0001-8253-272910 NOUN bracis-19045 1 139 & & CCONJ bracis-19045 1 140 michel michel PROPN bracis-19045 1 141 f. f. PROPN bracis-19045 1 142 c. c. PROPN bracis-19045 1 143 haddad haddad PROPN bracis-19045 1 144     SPACE bracis-19045 1 145 orcid orcid PROPN bracis-19045 1 146 : : PUNCT bracis-19045 1 147 orcid.org/0000-0002-0978-952511,12 orcid.org/0000-0002-0978-952511,12 VERB bracis-19045 1 148     SPACE bracis-19045 1 149 part part NOUN bracis-19045 1 150 of of ADP bracis-19045 1 151 the the DET bracis-19045 1 152 book book NOUN bracis-19045 1 153 series series NOUN bracis-19045 1 154 : : PUNCT bracis-19045 1 155 lecture lecture NOUN bracis-19045 1 156 notes note NOUN bracis-19045 1 157 in in ADP bracis-19045 1 158 computer computer NOUN bracis-19045 1 159 science science NOUN bracis-19045 1 160 ( ( PUNCT bracis-19045 1 161 ( ( PUNCT bracis-19045 1 162 lnai lnai ADJ bracis-19045 1 163 , , PUNCT bracis-19045 1 164 volume volume NOUN bracis-19045 1 165 13073 13073 NUM bracis-19045 1 166 ) ) PUNCT bracis-19045 1 167 ) ) PUNCT bracis-19045 1 168 included include VERB bracis-19045 1 169 in in ADP bracis-19045 1 170 the the DET bracis-19045 1 171 following follow VERB bracis-19045 1 172 conference conference NOUN bracis-19045 1 173 series series NOUN bracis-19045 1 174 : : PUNCT bracis-19045 1 175 brazilian brazilian ADJ bracis-19045 1 176 conference conference NOUN bracis-19045 1 177 on on ADP bracis-19045 1 178 intelligent intelligent ADJ bracis-19045 1 179 systems system NOUN bracis-19045 1 180 814 814 NUM bracis-19045 1 181 accesses access NOUN bracis-19045 1 182 abstract abstract ADJ bracis-19045 1 183 numerous numerous ADJ bracis-19045 1 184 problems problem NOUN bracis-19045 1 185 in in ADP bracis-19045 1 186 machine machine NOUN bracis-19045 1 187 learning learning NOUN bracis-19045 1 188 require require VERB bracis-19045 1 189 some some DET bracis-19045 1 190 type type NOUN bracis-19045 1 191 of of ADP bracis-19045 1 192 dimensionality dimensionality NOUN bracis-19045 1 193 reduction reduction NOUN bracis-19045 1 194 . . PUNCT bracis-19045 2 1 unsupervised unsupervised ADJ bracis-19045 2 2 metric metric ADJ bracis-19045 2 3 learning learning NOUN bracis-19045 2 4 deals deal NOUN bracis-19045 2 5 with with ADP bracis-19045 2 6 the the DET bracis-19045 2 7 definition definition NOUN bracis-19045 2 8 of of ADP bracis-19045 2 9 intrinsic intrinsic ADJ bracis-19045 2 10 and and CCONJ bracis-19045 2 11 adaptive adaptive ADJ bracis-19045 2 12 distance distance NOUN bracis-19045 2 13 functions function NOUN bracis-19045 2 14 of of ADP bracis-19045 2 15 a a DET bracis-19045 2 16 dataset dataset NOUN bracis-19045 2 17 . . PUNCT bracis-19045 3 1 locally locally ADV bracis-19045 3 2 linear linear ADJ bracis-19045 3 3 embedding embed VERB bracis-19045 3 4 ( ( PUNCT bracis-19045 3 5 lle lle PROPN bracis-19045 3 6 ) ) PUNCT bracis-19045 3 7 consists consist VERB bracis-19045 3 8 of of ADP bracis-19045 3 9 a a DET bracis-19045 3 10 widely widely ADV bracis-19045 3 11 used use VERB bracis-19045 3 12 manifold manifold ADJ bracis-19045 3 13 learning learning NOUN bracis-19045 3 14 algorithm algorithm NOUN bracis-19045 3 15 that that PRON bracis-19045 3 16 applies apply VERB bracis-19045 3 17 dimensionality dimensionality NOUN bracis-19045 3 18 reduction reduction NOUN bracis-19045 3 19 to to PART bracis-19045 3 20 find find VERB bracis-19045 3 21 a a DET bracis-19045 3 22 more more ADV bracis-19045 3 23 compact compact ADJ bracis-19045 3 24 and and CCONJ bracis-19045 3 25 meaningful meaningful ADJ bracis-19045 3 26 representation representation NOUN bracis-19045 3 27 of of ADP bracis-19045 3 28 the the DET bracis-19045 3 29 observed observe VERB bracis-19045 3 30 data datum NOUN bracis-19045 3 31 through through ADP bracis-19045 3 32 the the DET bracis-19045 3 33 capture capture NOUN bracis-19045 3 34 of of ADP bracis-19045 3 35 the the DET bracis-19045 3 36 local local ADJ bracis-19045 3 37 geometry geometry NOUN bracis-19045 3 38 of of ADP bracis-19045 3 39 the the DET bracis-19045 3 40 patches patch NOUN bracis-19045 3 41 . . PUNCT bracis-19045 4 1 in in ADP bracis-19045 4 2 order order NOUN bracis-19045 4 3 to to PART bracis-19045 4 4 overcome overcome VERB bracis-19045 4 5 relevant relevant ADJ bracis-19045 4 6 limitations limitation NOUN bracis-19045 4 7 of of ADP bracis-19045 4 8 the the DET bracis-19045 4 9 lle lle PROPN bracis-19045 4 10 approach approach NOUN bracis-19045 4 11 , , PUNCT bracis-19045 4 12 we we PRON bracis-19045 4 13 introduce introduce VERB bracis-19045 4 14 the the DET bracis-19045 4 15 lle lle PROPN bracis-19045 4 16 kullback kullback NOUN bracis-19045 4 17 - - PUNCT bracis-19045 4 18 leibler leibler NOUN bracis-19045 4 19 ( ( PUNCT bracis-19045 4 20 lle lle PROPN bracis-19045 4 21 - - PUNCT bracis-19045 4 22 kl kl NOUN bracis-19045 4 23 ) ) PUNCT bracis-19045 4 24 method method NOUN bracis-19045 4 25 . . PUNCT bracis-19045 5 1 our our PRON bracis-19045 5 2 objective objective NOUN bracis-19045 5 3 with with ADP bracis-19045 5 4 such such DET bracis-19045 5 5 a a DET bracis-19045 5 6 methodological methodological ADJ bracis-19045 5 7 modification modification NOUN bracis-19045 5 8 is be AUX bracis-19045 5 9 to to PART bracis-19045 5 10 increase increase VERB bracis-19045 5 11 the the DET bracis-19045 5 12 robustness robustness NOUN bracis-19045 5 13 of of ADP bracis-19045 5 14 the the DET bracis-19045 5 15 lle lle NOUN bracis-19045 5 16 to to ADP bracis-19045 5 17 the the DET bracis-19045 5 18 presence presence NOUN bracis-19045 5 19 of of ADP bracis-19045 5 20 noise noise NOUN bracis-19045 5 21 or or CCONJ bracis-19045 5 22 outliers outlier NOUN bracis-19045 5 23 in in ADP bracis-19045 5 24 the the DET bracis-19045 5 25 data datum NOUN bracis-19045 5 26 . . PUNCT bracis-19045 6 1 the the DET bracis-19045 6 2 proposed propose VERB bracis-19045 6 3 method method NOUN bracis-19045 6 4 employs employ VERB bracis-19045 6 5 the the DET bracis-19045 6 6 kl kl PROPN bracis-19045 6 7 divergence divergence NOUN bracis-19045 6 8 between between ADP bracis-19045 6 9 patches patch NOUN bracis-19045 6 10 of of ADP bracis-19045 6 11 the the DET bracis-19045 6 12 knn knn PROPN bracis-19045 6 13 graph graph NOUN bracis-19045 6 14 instead instead ADV bracis-19045 6 15 of of ADP bracis-19045 6 16 the the DET bracis-19045 6 17 pointwise pointwise ADJ bracis-19045 6 18 euclidean euclidean ADJ bracis-19045 6 19 metric metric NOUN bracis-19045 6 20 . . PUNCT bracis-19045 7 1 our our PRON bracis-19045 7 2 empirical empirical ADJ bracis-19045 7 3 results result NOUN bracis-19045 7 4 using use VERB bracis-19045 7 5 several several ADJ bracis-19045 7 6 real real ADJ bracis-19045 7 7 - - PUNCT bracis-19045 7 8 world world NOUN bracis-19045 7 9 datasets dataset NOUN bracis-19045 7 10 indicate indicate VERB bracis-19045 7 11 that that SCONJ bracis-19045 7 12 the the DET bracis-19045 7 13 proposed propose VERB bracis-19045 7 14 method method NOUN bracis-19045 7 15 delivers deliver VERB bracis-19045 7 16 a a DET bracis-19045 7 17 superior superior ADJ bracis-19045 7 18 clustering clustering ADJ bracis-19045 7 19 allocation allocation NOUN bracis-19045 7 20 compared compare VERB bracis-19045 7 21 to to ADP bracis-19045 7 22 state state NOUN bracis-19045 7 23 - - PUNCT bracis-19045 7 24 of of ADP bracis-19045 7 25 - - PUNCT bracis-19045 7 26 the the DET bracis-19045 7 27 - - PUNCT bracis-19045 7 28 art art NOUN bracis-19045 7 29 methods method NOUN bracis-19045 7 30 of of ADP bracis-19045 7 31 dimensionality dimensionality NOUN bracis-19045 7 32 reduction reduction NOUN bracis-19045 7 33 - - PUNCT bracis-19045 7 34 based base VERB bracis-19045 7 35 metric metric ADJ bracis-19045 7 36 learning learning NOUN bracis-19045 7 37 . . PUNCT bracis-19045 8 1 access access NOUN bracis-19045 8 2 provided provide VERB bracis-19045 8 3 by by ADP bracis-19045 8 4 university university PROPN bracis-19045 8 5 of of ADP bracis-19045 8 6 notre notre PROPN bracis-19045 8 7 dame dame PROPN bracis-19045 8 8 hesburgh hesburgh PROPN bracis-19045 8 9 library library PROPN bracis-19045 8 10 . . PUNCT bracis-19045 9 1 download download PROPN bracis-19045 9 2 conference conference NOUN bracis-19045 9 3 paper paper NOUN bracis-19045 9 4 pdf pdf NOUN bracis-19045 9 5 similar similar ADJ bracis-19045 9 6 content content NOUN bracis-19045 9 7 being be AUX bracis-19045 9 8 viewed view VERB bracis-19045 9 9 by by ADP bracis-19045 9 10 others other NOUN bracis-19045 9 11 using use VERB bracis-19045 9 12 kullback kullback NOUN bracis-19045 9 13 - - PUNCT bracis-19045 9 14 liebler liebler NOUN bracis-19045 9 15 divergence divergence NOUN bracis-19045 9 16 based base VERB bracis-19045 9 17 meta meta ADJ bracis-19045 9 18 - - PUNCT bracis-19045 9 19 learning learn VERB bracis-19045 9 20 algorithm algorithm NOUN bracis-19045 9 21 for for ADP bracis-19045 9 22 few few ADJ bracis-19045 9 23 - - PUNCT bracis-19045 9 24 shot shot NOUN bracis-19045 9 25 skin skin NOUN bracis-19045 9 26 cancer cancer NOUN bracis-19045 9 27 image image NOUN bracis-19045 9 28 classification classification NOUN bracis-19045 9 29 : : PUNCT bracis-19045 9 30 literature literature NOUN bracis-19045 9 31 review review NOUN bracis-19045 9 32 and and CCONJ bracis-19045 9 33 a a DET bracis-19045 9 34 conceptual conceptual ADJ bracis-19045 9 35 framework framework NOUN bracis-19045 9 36 chapter chapter NOUN bracis-19045 9 37 © © PROPN bracis-19045 9 38 2022 2022 NUM bracis-19045 9 39 dissimilarity dissimilarity NOUN bracis-19045 9 40 space space NOUN bracis-19045 9 41 reinforced reinforce VERB bracis-19045 9 42 with with ADP bracis-19045 9 43 manifold manifold ADJ bracis-19045 9 44 learning learning NOUN bracis-19045 9 45 and and CCONJ bracis-19045 9 46 latent latent NOUN bracis-19045 9 47 space space NOUN bracis-19045 9 48 modeling modeling NOUN bracis-19045 9 49 for for ADP bracis-19045 9 50 improved improved ADJ bracis-19045 9 51 pattern pattern NOUN bracis-19045 9 52 classification classification NOUN bracis-19045 9 53 article article NOUN bracis-19045 9 54 open open ADJ bracis-19045 9 55 access access NOUN bracis-19045 9 56 18 18 NUM bracis-19045 9 57 october october NOUN bracis-19045 9 58 2021 2021 NUM bracis-19045 9 59 dimensionality dimensionality NOUN bracis-19045 9 60 reduction reduction NOUN bracis-19045 9 61 and and CCONJ bracis-19045 9 62 visualization visualization NOUN bracis-19045 9 63 by by ADP bracis-19045 9 64 doubly doubly ADV bracis-19045 9 65 kernelized kernelize VERB bracis-19045 9 66 unit unit NOUN bracis-19045 9 67 ball ball NOUN bracis-19045 9 68 embedding embed VERB bracis-19045 9 69 chapter chapter NOUN bracis-19045 9 70 © © PROPN bracis-19045 9 71 2018 2018 NUM bracis-19045 9 72 explore explore VERB bracis-19045 9 73 related relate VERB bracis-19045 9 74 subjects subject NOUN bracis-19045 9 75 discover discover VERB bracis-19045 9 76 the the DET bracis-19045 9 77 latest late ADJ bracis-19045 9 78 articles article NOUN bracis-19045 9 79 , , PUNCT bracis-19045 9 80 books book NOUN bracis-19045 9 81 and and CCONJ bracis-19045 9 82 news news NOUN bracis-19045 9 83 in in ADP bracis-19045 9 84 related related ADJ bracis-19045 9 85 subjects subject NOUN bracis-19045 9 86 , , PUNCT bracis-19045 9 87 suggested suggest VERB bracis-19045 9 88 using use VERB bracis-19045 9 89 machine machine NOUN bracis-19045 9 90 learning learning NOUN bracis-19045 9 91 . . PUNCT bracis-19045 10 1 global global ADJ bracis-19045 10 2 analysis analysis NOUN bracis-19045 10 3 and and CCONJ bracis-19045 10 4 analysis analysis NOUN bracis-19045 10 5 on on ADP bracis-19045 10 6 manifolds manifold NOUN bracis-19045 10 7 learning learn VERB bracis-19045 10 8 algorithms algorithm NOUN bracis-19045 10 9 learning learn VERB bracis-19045 10 10 theory theory NOUN bracis-19045 10 11 statistical statistical ADJ bracis-19045 10 12 learning learning NOUN bracis-19045 10 13 diffusion diffusion NOUN bracis-19045 10 14 processes process NOUN bracis-19045 10 15 and and CCONJ bracis-19045 10 16 stochastic stochastic ADJ bracis-19045 10 17 analysis analysis NOUN bracis-19045 10 18 on on ADP bracis-19045 10 19 manifolds manifolds ADJ bracis-19045 10 20 statistical statistical ADJ bracis-19045 10 21 theory theory NOUN bracis-19045 10 22 and and CCONJ bracis-19045 10 23 methods method NOUN bracis-19045 10 24 1 1 NUM bracis-19045 10 25 introduction introduction NOUN bracis-19045 10 26 the the DET bracis-19045 10 27 main main ADJ bracis-19045 10 28 objective objective NOUN bracis-19045 10 29 of of ADP bracis-19045 10 30 dimensionality dimensionality NOUN bracis-19045 10 31 reduction reduction NOUN bracis-19045 10 32 unsupervised unsupervised ADJ bracis-19045 10 33 metric metric ADJ bracis-19045 10 34 learning learning NOUN bracis-19045 10 35 is be AUX bracis-19045 10 36 to to PART bracis-19045 10 37 produce produce VERB bracis-19045 10 38 a a DET bracis-19045 10 39 lower low ADJ bracis-19045 10 40 dimensional dimensional ADJ bracis-19045 10 41 representation representation NOUN bracis-19045 10 42 that that PRON bracis-19045 10 43 preserves preserve VERB bracis-19045 10 44 the the DET bracis-19045 10 45 intrinsic intrinsic ADJ bracis-19045 10 46 local local ADJ bracis-19045 10 47 geometry geometry NOUN bracis-19045 10 48 of of ADP bracis-19045 10 49 the the DET bracis-19045 10 50 data datum NOUN bracis-19045 10 51 . . PUNCT bracis-19045 11 1 the the DET bracis-19045 11 2 locally locally ADV bracis-19045 11 3 linear linear NOUN bracis-19045 11 4 embedding embed VERB bracis-19045 11 5 ( ( PUNCT bracis-19045 11 6 lle lle PROPN bracis-19045 11 7 ) ) PUNCT bracis-19045 11 8 consists consist VERB bracis-19045 11 9 of of ADP bracis-19045 11 10 one one NUM bracis-19045 11 11 of of ADP bracis-19045 11 12 the the DET bracis-19045 11 13 pioneering pioneering ADJ bracis-19045 11 14 algorithms algorithm NOUN bracis-19045 11 15 of of ADP bracis-19045 11 16 such such DET bracis-19045 11 17 a a DET bracis-19045 11 18 class class NOUN bracis-19045 11 19 , , PUNCT bracis-19045 11 20 which which PRON bracis-19045 11 21 has have AUX bracis-19045 11 22 been be AUX bracis-19045 11 23 successfully successfully ADV bracis-19045 11 24 applied apply VERB bracis-19045 11 25 to to ADP bracis-19045 11 26 non non ADJ bracis-19045 11 27 - - ADJ bracis-19045 11 28 linear linear ADJ bracis-19045 11 29 feature feature NOUN bracis-19045 11 30 extraction extraction NOUN bracis-19045 11 31 for for ADP bracis-19045 11 32 pattern pattern NOUN bracis-19045 11 33 classification classification NOUN bracis-19045 11 34 tasks task NOUN bracis-19045 11 35 [ [ X bracis-19045 11 36 9 9 NUM bracis-19045 11 37 ] ] PUNCT bracis-19045 11 38 . . PUNCT bracis-19045 12 1 although although SCONJ bracis-19045 12 2 more more ADV bracis-19045 12 3 efficient efficient ADJ bracis-19045 12 4 than than ADP bracis-19045 12 5 linear linear ADJ bracis-19045 12 6 methods method NOUN bracis-19045 12 7 , , PUNCT bracis-19045 12 8 the the DET bracis-19045 12 9 lle lle NOUN bracis-19045 12 10 still still ADV bracis-19045 12 11 has have VERB bracis-19045 12 12 some some DET bracis-19045 12 13 important important ADJ bracis-19045 12 14 limitations limitation NOUN bracis-19045 12 15 . . PUNCT bracis-19045 13 1 firstly firstly ADV bracis-19045 13 2 , , PUNCT bracis-19045 13 3 it it PRON bracis-19045 13 4 is be AUX bracis-19045 13 5 rather rather ADV bracis-19045 13 6 sensitive sensitive ADJ bracis-19045 13 7 to to ADP bracis-19045 13 8 noise noise NOUN bracis-19045 13 9 or or CCONJ bracis-19045 13 10 outliers outlier NOUN bracis-19045 13 11 [ [ X bracis-19045 13 12 14 14 NUM bracis-19045 13 13 ] ] PUNCT bracis-19045 13 14 . . PUNCT bracis-19045 14 1 secondly secondly ADV bracis-19045 14 2 , , PUNCT bracis-19045 14 3 in in ADP bracis-19045 14 4 datasets dataset NOUN bracis-19045 14 5 that that PRON bracis-19045 14 6 are be AUX bracis-19045 14 7 not not PART bracis-19045 14 8 represented represent VERB bracis-19045 14 9 by by ADP bracis-19045 14 10 smooth smooth ADJ bracis-19045 14 11 manifolds manifold NOUN bracis-19045 14 12 , , PUNCT bracis-19045 14 13 the the DET bracis-19045 14 14 lle lle NOUN bracis-19045 14 15 frequently frequently ADV bracis-19045 14 16 fails fail VERB bracis-19045 14 17 to to PART bracis-19045 14 18 produce produce VERB bracis-19045 14 19 reasonable reasonable ADJ bracis-19045 14 20 results result NOUN bracis-19045 14 21 . . PUNCT bracis-19045 15 1 it it PRON bracis-19045 15 2 is be AUX bracis-19045 15 3 worth worth ADJ bracis-19045 15 4 mentioning mention VERB bracis-19045 15 5 that that SCONJ bracis-19045 15 6 variations variation NOUN bracis-19045 15 7 of of ADP bracis-19045 15 8 the the DET bracis-19045 15 9 lle lle NOUN bracis-19045 15 10 have have AUX bracis-19045 15 11 been be AUX bracis-19045 15 12 proposed propose VERB bracis-19045 15 13 to to PART bracis-19045 15 14 overcome overcome VERB bracis-19045 15 15 such such ADJ bracis-19045 15 16 limitations limitation NOUN bracis-19045 15 17 , , PUNCT bracis-19045 15 18 with with ADP bracis-19045 15 19 varying vary VERB bracis-19045 15 20 degrees degree NOUN bracis-19045 15 21 of of ADP bracis-19045 15 22 success success NOUN bracis-19045 15 23 . . PUNCT bracis-19045 16 1 in in ADP bracis-19045 16 2 hessian hessian ADJ bracis-19045 16 3 eigenmaps eigenmap NOUN bracis-19045 16 4 , , PUNCT bracis-19045 16 5 the the DET bracis-19045 16 6 local local ADJ bracis-19045 16 7 covariance covariance NOUN bracis-19045 16 8 matrix matrix NOUN bracis-19045 16 9 used use VERB bracis-19045 16 10 in in ADP bracis-19045 16 11 the the DET bracis-19045 16 12 computation computation NOUN bracis-19045 16 13 of of ADP bracis-19045 16 14 the the DET bracis-19045 16 15 optimal optimal ADJ bracis-19045 16 16 reconstruction reconstruction NOUN bracis-19045 16 17 weights weight NOUN bracis-19045 16 18 is be AUX bracis-19045 16 19 replaced replace VERB bracis-19045 16 20 by by ADP bracis-19045 16 21 the the DET bracis-19045 16 22 hessian hessian ADJ bracis-19045 16 23 matrix matrix NOUN bracis-19045 16 24 ( ( PUNCT bracis-19045 16 25 i.e. i.e. X bracis-19045 16 26 , , PUNCT bracis-19045 16 27 the the DET bracis-19045 16 28 second second ADJ bracis-19045 16 29 order order NOUN bracis-19045 16 30 derivatives derivative NOUN bracis-19045 16 31 matrix matrix NOUN bracis-19045 16 32 ) ) PUNCT bracis-19045 16 33 , , PUNCT bracis-19045 16 34 which which PRON bracis-19045 16 35 encodes encode VERB bracis-19045 16 36 relevant relevant ADJ bracis-19045 16 37 curvature curvature NOUN bracis-19045 16 38 information information NOUN bracis-19045 16 39 [ [ X bracis-19045 16 40 1 1 NUM bracis-19045 16 41 , , PUNCT bracis-19045 16 42 13 13 NUM bracis-19045 16 43 , , PUNCT bracis-19045 16 44 15 15 NUM bracis-19045 16 45 ] ] PUNCT bracis-19045 16 46 . . PUNCT bracis-19045 17 1 a a DET bracis-19045 17 2 further further ADJ bracis-19045 17 3 extension extension NOUN bracis-19045 17 4 is be AUX bracis-19045 17 5 known know VERB bracis-19045 17 6 as as ADP bracis-19045 17 7 the the DET bracis-19045 17 8 local local ADJ bracis-19045 17 9 tangent tangent NOUN bracis-19045 17 10 space space NOUN bracis-19045 17 11 alignment alignment NOUN bracis-19045 17 12 ( ( PUNCT bracis-19045 17 13 ltsa ltsa PROPN bracis-19045 17 14 ) ) PUNCT bracis-19045 17 15 . . PUNCT bracis-19045 18 1 the the DET bracis-19045 18 2 difference difference NOUN bracis-19045 18 3 is be AUX bracis-19045 18 4 that that SCONJ bracis-19045 18 5 the the DET bracis-19045 18 6 ltsa ltsa NOUN bracis-19045 18 7 builds build VERB bracis-19045 18 8 the the DET bracis-19045 18 9 locally locally ADV bracis-19045 18 10 linear linear ADJ bracis-19045 18 11 patch patch NOUN bracis-19045 18 12 with with ADP bracis-19045 18 13 an an DET bracis-19045 18 14 approximation approximation NOUN bracis-19045 18 15 of of ADP bracis-19045 18 16 the the DET bracis-19045 18 17 tangent tangent ADJ bracis-19045 18 18 space space NOUN bracis-19045 18 19 through through ADP bracis-19045 18 20 the the DET bracis-19045 18 21 application application NOUN bracis-19045 18 22 of of ADP bracis-19045 18 23 the the DET bracis-19045 18 24 principal principal ADJ bracis-19045 18 25 component component NOUN bracis-19045 18 26 analysis analysis NOUN bracis-19045 18 27 ( ( PUNCT bracis-19045 18 28 pca pca NOUN bracis-19045 18 29 ) ) PUNCT bracis-19045 18 30 over over ADP bracis-19045 18 31 a a DET bracis-19045 18 32 linear linear ADJ bracis-19045 18 33 patch patch NOUN bracis-19045 18 34 . . PUNCT bracis-19045 19 1 subsequently subsequently ADV bracis-19045 19 2 , , PUNCT bracis-19045 19 3 the the DET bracis-19045 19 4 local local ADJ bracis-19045 19 5 representations representation NOUN bracis-19045 19 6 are be AUX bracis-19045 19 7 aligned align VERB bracis-19045 19 8 to to ADP bracis-19045 19 9 the the DET bracis-19045 19 10 point point NOUN bracis-19045 19 11 in in ADP bracis-19045 19 12 which which PRON bracis-19045 19 13 all all DET bracis-19045 19 14 tangent tangent NOUN bracis-19045 19 15 spaces space NOUN bracis-19045 19 16 become become VERB bracis-19045 19 17 aligned aligned ADJ bracis-19045 19 18 in in ADP bracis-19045 19 19 the the DET bracis-19045 19 20 unfolded unfold VERB bracis-19045 19 21 manifold manifold NOUN bracis-19045 19 22 [ [ X bracis-19045 19 23 16 16 NUM bracis-19045 19 24 ] ] PUNCT bracis-19045 19 25 . . PUNCT bracis-19045 20 1 in in ADP bracis-19045 20 2 addition addition NOUN bracis-19045 20 3 , , PUNCT bracis-19045 20 4 the the DET bracis-19045 20 5 modified modified ADJ bracis-19045 20 6 lle lle NOUN bracis-19045 20 7 ( ( PUNCT bracis-19045 20 8 mlle mlle NOUN bracis-19045 20 9 ) ) PUNCT bracis-19045 20 10 introduces introduce VERB bracis-19045 20 11 several several ADJ bracis-19045 20 12 linearly linearly ADV bracis-19045 20 13 independent independent ADJ bracis-19045 20 14 weight weight NOUN bracis-19045 20 15 vectors vector NOUN bracis-19045 20 16 for for ADP bracis-19045 20 17 each each DET bracis-19045 20 18 neighborhood neighborhood NOUN bracis-19045 20 19 of of ADP bracis-19045 20 20 the the DET bracis-19045 20 21 knn knn NOUN bracis-19045 20 22 graph graph NOUN bracis-19045 20 23 . . PUNCT bracis-19045 21 1 some some DET bracis-19045 21 2 works work VERB bracis-19045 21 3 in in ADP bracis-19045 21 4 the the DET bracis-19045 21 5 relevant relevant ADJ bracis-19045 21 6 literature literature NOUN bracis-19045 21 7 describe describe VERB bracis-19045 21 8 that that SCONJ bracis-19045 21 9 the the DET bracis-19045 21 10 local local ADJ bracis-19045 21 11 geometry geometry NOUN bracis-19045 21 12 of of ADP bracis-19045 21 13 mlle mlle NOUN bracis-19045 21 14 is be AUX bracis-19045 21 15 more more ADV bracis-19045 21 16 stable stable ADJ bracis-19045 21 17 in in ADP bracis-19045 21 18 comparison comparison NOUN bracis-19045 21 19 with with ADP bracis-19045 21 20 the the DET bracis-19045 21 21 lle lle NOUN bracis-19045 22 1 [ [ X bracis-19045 22 2 17 17 NUM bracis-19045 22 3 ] ] PUNCT bracis-19045 22 4 . . PUNCT bracis-19045 23 1 one one NUM bracis-19045 23 2 remaining remain VERB bracis-19045 23 3 issue issue NOUN bracis-19045 23 4 still still ADV bracis-19045 23 5 present present ADJ bracis-19045 23 6 in in ADP bracis-19045 23 7 such such ADJ bracis-19045 23 8 recent recent ADJ bracis-19045 23 9 developments development NOUN bracis-19045 23 10 of of ADP bracis-19045 23 11 the the DET bracis-19045 23 12 lle lle PROPN bracis-19045 23 13 method method NOUN bracis-19045 23 14 is be AUX bracis-19045 23 15 the the DET bracis-19045 23 16 adoption adoption NOUN bracis-19045 23 17 of of ADP bracis-19045 23 18 euclidean euclidean ADJ bracis-19045 23 19 distance distance NOUN bracis-19045 23 20 as as ADP bracis-19045 23 21 the the DET bracis-19045 23 22 similarity similarity NOUN bracis-19045 23 23 measure measure NOUN bracis-19045 23 24 within within ADP bracis-19045 23 25 a a DET bracis-19045 23 26 linear linear ADJ bracis-19045 23 27 patch patch NOUN bracis-19045 23 28 . . PUNCT bracis-19045 24 1 in in ADP bracis-19045 24 2 the the DET bracis-19045 24 3 proposed propose VERB bracis-19045 24 4 lle lle PROPN bracis-19045 24 5 - - PUNCT bracis-19045 24 6 kl kl NOUN bracis-19045 24 7 method method NOUN bracis-19045 24 8 , , PUNCT bracis-19045 24 9 we we PRON bracis-19045 24 10 incorporate incorporate VERB bracis-19045 24 11 the the DET bracis-19045 24 12 kl kl PROPN bracis-19045 24 13 divergence divergence NOUN bracis-19045 24 14 into into ADP bracis-19045 24 15 the the DET bracis-19045 24 16 estimation estimation NOUN bracis-19045 24 17 of of ADP bracis-19045 24 18 the the DET bracis-19045 24 19 optimal optimal ADJ bracis-19045 24 20 reconstruction reconstruction NOUN bracis-19045 24 21 weights weight NOUN bracis-19045 24 22 . . PUNCT bracis-19045 25 1 the the DET bracis-19045 25 2 main main ADJ bracis-19045 25 3 contribution contribution NOUN bracis-19045 25 4 of of ADP bracis-19045 25 5 the the DET bracis-19045 25 6 proposed propose VERB bracis-19045 25 7 method method NOUN bracis-19045 25 8 is be AUX bracis-19045 25 9 that that SCONJ bracis-19045 25 10 , , PUNCT bracis-19045 25 11 through through ADP bracis-19045 25 12 the the DET bracis-19045 25 13 replacement replacement NOUN bracis-19045 25 14 of of ADP bracis-19045 25 15 the the DET bracis-19045 25 16 pointwise pointwise ADJ bracis-19045 25 17 euclidean euclidean ADJ bracis-19045 25 18 distance distance NOUN bracis-19045 25 19 by by ADP bracis-19045 25 20 a a DET bracis-19045 25 21 patch patch NOUN bracis-19045 25 22 - - PUNCT bracis-19045 25 23 based base VERB bracis-19045 25 24 information information NOUN bracis-19045 25 25 - - PUNCT bracis-19045 25 26 theoretic theoretic NOUN bracis-19045 25 27 distance distance NOUN bracis-19045 25 28 ( ( PUNCT bracis-19045 25 29 i.e. i.e. X bracis-19045 25 30 , , PUNCT bracis-19045 25 31 kl kl NOUN bracis-19045 25 32 divergence divergence NOUN bracis-19045 25 33 ) ) PUNCT bracis-19045 25 34 , , PUNCT bracis-19045 25 35 the the DET bracis-19045 25 36 lle lle NOUN bracis-19045 25 37 becomes become VERB bracis-19045 25 38 less less ADV bracis-19045 25 39 sensitive sensitive ADJ bracis-19045 25 40 to to ADP bracis-19045 25 41 the the DET bracis-19045 25 42 presence presence NOUN bracis-19045 25 43 of of ADP bracis-19045 25 44 noise noise NOUN bracis-19045 25 45 or or CCONJ bracis-19045 25 46 outliers outlier NOUN bracis-19045 25 47 . . PUNCT bracis-19045 26 1 our our PRON bracis-19045 26 2 empirical empirical ADJ bracis-19045 26 3 clustering clustering NOUN bracis-19045 26 4 analysis analysis NOUN bracis-19045 26 5 performed perform VERB bracis-19045 26 6 subsequently subsequently ADV bracis-19045 26 7 to to ADP bracis-19045 26 8 the the DET bracis-19045 26 9 dimensionality dimensionality NOUN bracis-19045 26 10 reduction reduction NOUN bracis-19045 26 11 for for ADP bracis-19045 26 12 several several ADJ bracis-19045 26 13 real real ADJ bracis-19045 26 14 - - PUNCT bracis-19045 26 15 world world NOUN bracis-19045 26 16 datasets dataset NOUN bracis-19045 26 17 popularly popularly ADV bracis-19045 26 18 used use VERB bracis-19045 26 19 in in ADP bracis-19045 26 20 the the DET bracis-19045 26 21 machine machine NOUN bracis-19045 26 22 learning learn VERB bracis-19045 26 23 literature literature NOUN bracis-19045 26 24 , , PUNCT bracis-19045 26 25 indicate indicate VERB bracis-19045 26 26 that that SCONJ bracis-19045 26 27 the the DET bracis-19045 26 28 proposed propose VERB bracis-19045 26 29 lle lle PROPN bracis-19045 26 30 - - PUNCT bracis-19045 26 31 kl kl PROPN bracis-19045 26 32 method method NOUN bracis-19045 26 33 is be AUX bracis-19045 26 34 capable capable ADJ bracis-19045 26 35 of of ADP bracis-19045 26 36 generating generate VERB bracis-19045 26 37 more more ADV bracis-19045 26 38 reasonable reasonable ADJ bracis-19045 26 39 clusters cluster NOUN bracis-19045 26 40 in in ADP bracis-19045 26 41 terms term NOUN bracis-19045 26 42 of of ADP bracis-19045 26 43 silhouette silhouette NOUN bracis-19045 26 44 coefficients coefficient NOUN bracis-19045 26 45 compared compare VERB bracis-19045 26 46 to to ADP bracis-19045 26 47 state state NOUN bracis-19045 26 48 - - PUNCT bracis-19045 26 49 of of ADP bracis-19045 26 50 - - PUNCT bracis-19045 26 51 the the DET bracis-19045 26 52 - - PUNCT bracis-19045 26 53 art art NOUN bracis-19045 26 54 manifold manifold ADJ bracis-19045 26 55 learning learning NOUN bracis-19045 26 56 algorithms algorithm NOUN bracis-19045 26 57 , , PUNCT bracis-19045 26 58 such such ADJ bracis-19045 26 59 as as ADP bracis-19045 26 60 the the DET bracis-19045 26 61 isomap isomap NOUN bracis-19045 27 1 [ [ X bracis-19045 27 2 12 12 NUM bracis-19045 27 3 ] ] PUNCT bracis-19045 27 4 and and CCONJ bracis-19045 27 5 umap umap ADJ bracis-19045 27 6 [ [ X bracis-19045 27 7 6 6 NUM bracis-19045 27 8 ] ] PUNCT bracis-19045 27 9 . . PUNCT bracis-19045 28 1 the the DET bracis-19045 28 2 remainder remainder NOUN bracis-19045 28 3 of of ADP bracis-19045 28 4 the the DET bracis-19045 28 5 paper paper NOUN bracis-19045 28 6 is be AUX bracis-19045 28 7 organized organize VERB bracis-19045 28 8 as as SCONJ bracis-19045 28 9 follows follow VERB bracis-19045 28 10 . . PUNCT bracis-19045 29 1 section section NOUN bracis-19045 29 2     SPACE bracis-19045 29 3 2 2 NUM bracis-19045 29 4 discusses discuss VERB bracis-19045 29 5 previous previous ADJ bracis-19045 29 6 relevant relevant ADJ bracis-19045 29 7 work work NOUN bracis-19045 29 8 , , PUNCT bracis-19045 29 9 with with ADP bracis-19045 29 10 focus focus NOUN bracis-19045 29 11 on on ADP bracis-19045 29 12 the the DET bracis-19045 29 13 relative relative ADJ bracis-19045 29 14 entropy entropy NOUN bracis-19045 29 15 and and CCONJ bracis-19045 29 16 the the DET bracis-19045 29 17 regular regular ADJ bracis-19045 29 18 lle lle PROPN bracis-19045 29 19 algorithm algorithm NOUN bracis-19045 29 20 . . PUNCT bracis-19045 30 1 section section NOUN bracis-19045 30 2     SPACE bracis-19045 30 3 3 3 NUM bracis-19045 30 4 details detail NOUN bracis-19045 30 5 the the DET bracis-19045 30 6 proposed propose VERB bracis-19045 30 7 lle lle PROPN bracis-19045 30 8 - - PUNCT bracis-19045 30 9 kl kl NOUN bracis-19045 30 10 algorithm algorithm NOUN bracis-19045 30 11 . . PUNCT bracis-19045 31 1 section section NOUN bracis-19045 31 2     SPACE bracis-19045 31 3 4 4 NUM bracis-19045 31 4 presents present VERB bracis-19045 31 5 the the DET bracis-19045 31 6 data datum NOUN bracis-19045 31 7 used use VERB bracis-19045 31 8 as as ADV bracis-19045 31 9 well well ADV bracis-19045 31 10 as as ADP bracis-19045 31 11 computational computational ADJ bracis-19045 31 12 experiments experiment NOUN bracis-19045 31 13 and and CCONJ bracis-19045 31 14 empirical empirical ADJ bracis-19045 31 15 results result NOUN bracis-19045 31 16 . . PUNCT bracis-19045 32 1 section section NOUN bracis-19045 32 2     SPACE bracis-19045 32 3 5 5 NUM bracis-19045 32 4 concludes conclude VERB bracis-19045 32 5 with with ADP bracis-19045 32 6 our our PRON bracis-19045 32 7 final final ADJ bracis-19045 32 8 remarks remark NOUN bracis-19045 32 9 and and CCONJ bracis-19045 32 10 suggestions suggestion NOUN bracis-19045 32 11 for for ADP bracis-19045 32 12 future future ADJ bracis-19045 32 13 research research NOUN bracis-19045 32 14 . . PUNCT bracis-19045 33 1 2 2 NUM bracis-19045 33 2 related relate VERB bracis-19045 33 3 work work NOUN bracis-19045 33 4 in in ADP bracis-19045 33 5 this this DET bracis-19045 33 6 section section NOUN bracis-19045 33 7 , , PUNCT bracis-19045 33 8 we we PRON bracis-19045 33 9 discuss discuss VERB bracis-19045 33 10 the the DET bracis-19045 33 11 relative relative ADJ bracis-19045 33 12 entropy entropy NOUN bracis-19045 33 13 ( ( PUNCT bracis-19045 33 14 kl kl NOUN bracis-19045 33 15 divergence divergence NOUN bracis-19045 33 16 ) ) PUNCT bracis-19045 33 17 between between ADP bracis-19045 33 18 probability probability NOUN bracis-19045 33 19 density density NOUN bracis-19045 33 20 functions function NOUN bracis-19045 33 21 as as ADP bracis-19045 33 22 a a DET bracis-19045 33 23 similarity similarity NOUN bracis-19045 33 24 measure measure NOUN bracis-19045 33 25 among among ADP bracis-19045 33 26 random random ADJ bracis-19045 33 27 variables variable NOUN bracis-19045 33 28 , , PUNCT bracis-19045 33 29 exploring explore VERB bracis-19045 33 30 its its PRON bracis-19045 33 31 computation computation NOUN bracis-19045 33 32 in in ADP bracis-19045 33 33 the the DET bracis-19045 33 34 gaussian gaussian ADJ bracis-19045 33 35 case case NOUN bracis-19045 33 36 . . PUNCT bracis-19045 34 1 moreover moreover ADV bracis-19045 34 2 , , PUNCT bracis-19045 34 3 we we PRON bracis-19045 34 4 discuss discuss VERB bracis-19045 34 5 the the DET bracis-19045 34 6 lle lle PROPN bracis-19045 34 7 algorithm algorithm NOUN bracis-19045 34 8 and and CCONJ bracis-19045 34 9 detail detail VERB bracis-19045 34 10 its its PRON bracis-19045 34 11 mathematical mathematical ADJ bracis-19045 34 12 derivation derivation NOUN bracis-19045 34 13 . . PUNCT bracis-19045 35 1 2.1 2.1 NUM bracis-19045 35 2 kullback kullback NOUN bracis-19045 35 3 - - PUNCT bracis-19045 35 4 leibler leibler NOUN bracis-19045 35 5 divergence divergence NOUN bracis-19045 35 6 in in ADP bracis-19045 35 7 machine machine NOUN bracis-19045 35 8 learning learn VERB bracis-19045 35 9 applications application NOUN bracis-19045 35 10 , , PUNCT bracis-19045 35 11 the the DET bracis-19045 35 12 problem problem NOUN bracis-19045 35 13 of of ADP bracis-19045 35 14 quantifying quantify VERB bracis-19045 35 15 similarity similarity NOUN bracis-19045 35 16 levels level NOUN bracis-19045 35 17 between between ADP bracis-19045 35 18 different different ADJ bracis-19045 35 19 objects object NOUN bracis-19045 35 20 or or CCONJ bracis-19045 35 21 clusters cluster NOUN bracis-19045 35 22 consists consist VERB bracis-19045 35 23 of of ADP bracis-19045 35 24 a a DET bracis-19045 35 25 challenging challenging ADJ bracis-19045 35 26 task task NOUN bracis-19045 35 27 , , PUNCT bracis-19045 35 28 especially especially ADV bracis-19045 35 29 in in ADP bracis-19045 35 30 cases case NOUN bracis-19045 35 31 where where SCONJ bracis-19045 35 32 the the DET bracis-19045 35 33 standard standard ADJ bracis-19045 35 34 euclidean euclidean ADJ bracis-19045 35 35 distance distance NOUN bracis-19045 35 36 is be AUX bracis-19045 35 37 not not PART bracis-19045 35 38 a a DET bracis-19045 35 39 reasonable reasonable ADJ bracis-19045 35 40 choice choice NOUN bracis-19045 35 41 . . PUNCT bracis-19045 36 1 many many ADJ bracis-19045 36 2 studies study NOUN bracis-19045 36 3 on on ADP bracis-19045 36 4 feature feature NOUN bracis-19045 36 5 selection selection NOUN bracis-19045 36 6 adopt adopt VERB bracis-19045 36 7 statistical statistical ADJ bracis-19045 36 8 divergences divergence NOUN bracis-19045 36 9 to to PART bracis-19045 36 10 select select VERB bracis-19045 36 11 the the DET bracis-19045 36 12 set set NOUN bracis-19045 36 13 of of ADP bracis-19045 36 14 features feature NOUN bracis-19045 36 15 that that PRON bracis-19045 36 16 should should AUX bracis-19045 36 17 maximize maximize VERB bracis-19045 36 18 some some DET bracis-19045 36 19 measure measure NOUN bracis-19045 36 20 of of ADP bracis-19045 36 21 separation separation NOUN bracis-19045 36 22 between between ADP bracis-19045 36 23 classes class NOUN bracis-19045 36 24 . . PUNCT bracis-19045 37 1 part part NOUN bracis-19045 37 2 of of ADP bracis-19045 37 3 their their PRON bracis-19045 37 4 success success NOUN bracis-19045 37 5 is be AUX bracis-19045 37 6 due due ADJ bracis-19045 37 7 to to ADP bracis-19045 37 8 the the DET bracis-19045 37 9 fact fact NOUN bracis-19045 37 10 that that SCONJ bracis-19045 37 11 most most ADJ bracis-19045 37 12 dissimilarity dissimilarity NOUN bracis-19045 37 13 measures measure NOUN bracis-19045 37 14 are be AUX bracis-19045 37 15 related relate VERB bracis-19045 37 16 to to ADP bracis-19045 37 17 distance distance NOUN bracis-19045 37 18 metrics metric NOUN bracis-19045 37 19 . . PUNCT bracis-19045 38 1 in in ADP bracis-19045 38 2 such such DET bracis-19045 38 3 a a DET bracis-19045 38 4 context context NOUN bracis-19045 38 5 , , PUNCT bracis-19045 38 6 information information NOUN bracis-19045 38 7 theory theory NOUN bracis-19045 38 8 provides provide VERB bracis-19045 38 9 a a DET bracis-19045 38 10 solid solid ADJ bracis-19045 38 11 mathematical mathematical ADJ bracis-19045 38 12 background background NOUN bracis-19045 38 13 for for ADP bracis-19045 38 14 metric metric ADJ bracis-19045 38 15 learning learning NOUN bracis-19045 38 16 in in ADP bracis-19045 38 17 pattern pattern NOUN bracis-19045 38 18 classification classification NOUN bracis-19045 38 19 . . PUNCT bracis-19045 39 1 the the DET bracis-19045 39 2 entropy entropy NOUN bracis-19045 39 3 of of ADP bracis-19045 39 4 a a DET bracis-19045 39 5 continuous continuous ADJ bracis-19045 39 6 random random ADJ bracis-19045 39 7 vector vector NOUN bracis-19045 39 8 x x VERB bracis-19045 39 9 is be AUX bracis-19045 39 10 given give VERB bracis-19045 39 11 by by ADP bracis-19045 39 12 : : PUNCT bracis-19045 39 13 $ $ SYM bracis-19045 39 14 $ $ SYM bracis-19045 39 15 \begin{aligned \begin{aligne VERB bracis-19045 39 16 } } PUNCT bracis-19045 39 17 h(p h(p NOUN bracis-19045 39 18 ) ) PUNCT bracis-19045 39 19 = = SYM bracis-19045 39 20 \int \int NOUN bracis-19045 39 21 p(x p(x PROPN bracis-19045 39 22 ) ) PUNCT bracis-19045 40 1 [ [ X bracis-19045 40 2 log log NOUN bracis-19045 40 3 ~ ~ SYM bracis-19045 40 4 p(x p(x NOUN bracis-19045 40 5 ) ) PUNCT bracis-19045 40 6 ] ] PUNCT bracis-19045 40 7 dx dx PROPN bracis-19045 41 1 = = SYM bracis-19045 41 2 e e PROPN bracis-19045 41 3 \left \left PROPN bracis-19045 41 4 [ [ PUNCT bracis-19045 41 5 log log NOUN bracis-19045 41 6 ~ ~ SYM bracis-19045 41 7 p(x p(x NOUN bracis-19045 41 8 ) ) PUNCT bracis-19045 41 9 \right \right PROPN bracis-19045 41 10 ] ] X bracis-19045 41 11 \end{aligned}$$ \end{aligned}$$ X bracis-19045 41 12 ( ( PUNCT bracis-19045 41 13 1 1 NUM bracis-19045 41 14 ) ) PUNCT bracis-19045 41 15 where where SCONJ bracis-19045 41 16 p(x p(x NOUN bracis-19045 41 17 ) ) PUNCT bracis-19045 41 18 is be AUX bracis-19045 41 19 the the DET bracis-19045 41 20 probability probability NOUN bracis-19045 41 21 density density NOUN bracis-19045 41 22 function function NOUN bracis-19045 41 23 ( ( PUNCT bracis-19045 41 24 pdf pdf NOUN bracis-19045 41 25 ) ) PUNCT bracis-19045 41 26 . . PUNCT bracis-19045 42 1 in in ADP bracis-19045 42 2 a a DET bracis-19045 42 3 similar similar ADJ bracis-19045 42 4 fashion fashion NOUN bracis-19045 42 5 , , PUNCT bracis-19045 42 6 we we PRON bracis-19045 42 7 may may AUX bracis-19045 42 8 define define VERB bracis-19045 42 9 the the DET bracis-19045 42 10 cross cross NOUN bracis-19045 42 11 - - NOUN bracis-19045 42 12 entropy entropy NOUN bracis-19045 42 13 between between ADP bracis-19045 42 14 two two NUM bracis-19045 42 15 probability probability NOUN bracis-19045 42 16 density density NOUN bracis-19045 42 17 functions function NOUN bracis-19045 42 18 p(x p(x NOUN bracis-19045 42 19 ) ) PUNCT bracis-19045 42 20 and and CCONJ bracis-19045 42 21 q(x q(x NOUN bracis-19045 42 22 ): ): PUNCT bracis-19045 42 23 $ $ SYM bracis-19045 42 24 $ $ SYM bracis-19045 42 25 \begin{aligned \begin{aligne VERB bracis-19045 42 26 } } PUNCT bracis-19045 42 27 h(p h(p NOUN bracis-19045 42 28 , , PUNCT bracis-19045 42 29 q q X bracis-19045 42 30 ) ) PUNCT bracis-19045 42 31 = = SYM bracis-19045 42 32 \int \int NOUN bracis-19045 42 33 p(x p(x PROPN bracis-19045 42 34 ) ) PUNCT bracis-19045 43 1 [ [ X bracis-19045 43 2 log log NOUN bracis-19045 43 3 ~ ~ SYM bracis-19045 43 4 q(y q(y NOUN bracis-19045 43 5 ) ) PUNCT bracis-19045 43 6 ] ] PUNCT bracis-19045 43 7 dx dx PROPN bracis-19045 43 8 \end{aligned}$$ \end{aligned}$$ PROPN bracis-19045 43 9 ( ( PUNCT bracis-19045 43 10 2 2 NUM bracis-19045 43 11 ) ) PUNCT bracis-19045 43 12 the the DET bracis-19045 43 13 kl kl PROPN bracis-19045 43 14 divergence divergence NOUN bracis-19045 43 15 is be AUX bracis-19045 43 16 the the DET bracis-19045 43 17 difference difference NOUN bracis-19045 43 18 between between ADP bracis-19045 43 19 cross cross NOUN bracis-19045 43 20 - - NOUN bracis-19045 43 21 entropy entropy NOUN bracis-19045 43 22 and and CCONJ bracis-19045 43 23 entropy entropy NOUN bracis-19045 43 24 [ [ X bracis-19045 43 25 4 4 NUM bracis-19045 43 26 ] ] NOUN bracis-19045 43 27 : : PUNCT bracis-19045 43 28 $ $ SYM bracis-19045 43 29 $ $ SYM bracis-19045 43 30 \begin{aligned \begin{aligne VERB bracis-19045 43 31 } } PUNCT bracis-19045 43 32 d_{kl}(p d_{kl}(p NOUN bracis-19045 43 33 , , PUNCT bracis-19045 43 34 q q X bracis-19045 43 35 ) ) PUNCT bracis-19045 43 36 = = SYM bracis-19045 43 37 h(p h(p NOUN bracis-19045 43 38 , , PUNCT bracis-19045 43 39 q q NOUN bracis-19045 43 40 ) ) PUNCT bracis-19045 43 41 h(p h(p NOUN bracis-19045 43 42 ) ) PUNCT bracis-19045 43 43 = = SYM bracis-19045 43 44 \int \int PROPN bracis-19045 43 45 p(x p(x PROPN bracis-19045 43 46 ) ) PUNCT bracis-19045 43 47 log log NOUN bracis-19045 43 48 \left \left PROPN bracis-19045 43 49 ( ( PUNCT bracis-19045 43 50 \frac{p(x)}{q(x)}\right \frac{p(x)}{q(x)}\right PROPN bracis-19045 43 51 ) ) PUNCT bracis-19045 43 52 dx dx PROPN bracis-19045 43 53 = = SYM bracis-19045 43 54 e_{p e_{p PROPN bracis-19045 43 55 } } PUNCT bracis-19045 43 56 \left \left PROPN bracis-19045 43 57 [ [ PUNCT bracis-19045 43 58 log log NOUN bracis-19045 43 59 \left \left PROPN bracis-19045 43 60 ( ( PUNCT bracis-19045 43 61 \frac{p(x)}{q(x \frac{p(x)}{q(x NUM bracis-19045 43 62 ) ) PUNCT bracis-19045 43 63 } } PUNCT bracis-19045 44 1 \right \right PROPN bracis-19045 44 2 ) ) PUNCT bracis-19045 44 3 \right \right PROPN bracis-19045 44 4 ] ] PUNCT bracis-19045 44 5 \end{aligned}$$ \end{aligned}$$ X bracis-19045 44 6 ( ( PUNCT bracis-19045 44 7 3 3 X bracis-19045 44 8 ) ) PUNCT bracis-19045 44 9 an an DET bracis-19045 44 10 important important ADJ bracis-19045 44 11 property property NOUN bracis-19045 44 12 is be AUX bracis-19045 44 13 that that SCONJ bracis-19045 44 14 the the DET bracis-19045 44 15 relative relative ADJ bracis-19045 44 16 entropy entropy NOUN bracis-19045 44 17 is be AUX bracis-19045 44 18 non non ADJ bracis-19045 44 19 - - ADJ bracis-19045 44 20 negative negative ADJ bracis-19045 44 21 , , PUNCT bracis-19045 44 22 therefore therefore ADV bracis-19045 44 23 , , PUNCT bracis-19045 44 24 \(d_{kl}(p \(d_{kl}(p NOUN bracis-19045 44 25 , , PUNCT bracis-19045 44 26 q q X bracis-19045 44 27 ) ) PUNCT bracis-19045 44 28 \ge \ge PROPN bracis-19045 44 29 0\ 0\ PROPN bracis-19045 44 30 ) ) PUNCT bracis-19045 44 31 . . PUNCT bracis-19045 45 1 2.2 2.2 NUM bracis-19045 45 2 the the DET bracis-19045 45 3 lle lle PROPN bracis-19045 45 4 algorithm algorithm NOUN bracis-19045 45 5 the the DET bracis-19045 45 6 lle lle PROPN bracis-19045 45 7 consists consist VERB bracis-19045 45 8 of of ADP bracis-19045 45 9 a a DET bracis-19045 45 10 local local ADJ bracis-19045 45 11 method method NOUN bracis-19045 45 12 , , PUNCT bracis-19045 45 13 thus thus ADV bracis-19045 45 14 , , PUNCT bracis-19045 45 15 the the DET bracis-19045 45 16 new new ADJ bracis-19045 45 17 coordinates coordinate NOUN bracis-19045 45 18 of of ADP bracis-19045 45 19 any any DET bracis-19045 45 20 \(\vec \(\vec NOUN bracis-19045 45 21 { { PUNCT bracis-19045 45 22 x}_i x}_i PROPN bracis-19045 45 23 \in \in PROPN bracis-19045 45 24 r^m\ r^m\ PROPN bracis-19045 45 25 ) ) PUNCT bracis-19045 45 26 depends depend VERB bracis-19045 45 27 only only ADV bracis-19045 45 28 on on ADP bracis-19045 45 29 the the DET bracis-19045 45 30 neighborhood neighborhood NOUN bracis-19045 45 31 of of ADP bracis-19045 45 32 the the DET bracis-19045 45 33 respective respective ADJ bracis-19045 45 34 point point NOUN bracis-19045 45 35 . . PUNCT bracis-19045 46 1 the the DET bracis-19045 46 2 main main ADJ bracis-19045 46 3 hypothesis hypothesis NOUN bracis-19045 46 4 behind behind ADP bracis-19045 46 5 the the DET bracis-19045 46 6 lle lle NOUN bracis-19045 46 7 is be AUX bracis-19045 46 8 that that PRON bracis-19045 46 9 for for ADP bracis-19045 46 10 a a DET bracis-19045 46 11 sufficiently sufficiently ADV bracis-19045 46 12 high high ADJ bracis-19045 46 13 density density NOUN bracis-19045 46 14 of of ADP bracis-19045 46 15 samples sample NOUN bracis-19045 46 16 , , PUNCT bracis-19045 46 17 it it PRON bracis-19045 46 18 is be AUX bracis-19045 46 19 expected expect VERB bracis-19045 46 20 that that SCONJ bracis-19045 46 21 a a DET bracis-19045 46 22 vector vector NOUN bracis-19045 46 23 \(\vec \(\vec NOUN bracis-19045 46 24 { { PUNCT bracis-19045 46 25 x}_i\ x}_i\ NOUN bracis-19045 46 26 ) ) PUNCT bracis-19045 46 27 and and CCONJ bracis-19045 46 28 its its PRON bracis-19045 46 29 neighbors neighbor NOUN bracis-19045 46 30 define define VERB bracis-19045 46 31 a a DET bracis-19045 46 32 linear linear ADJ bracis-19045 46 33 patch patch NOUN bracis-19045 46 34 ( ( PUNCT bracis-19045 46 35 i.e. i.e. X bracis-19045 46 36 , , PUNCT bracis-19045 46 37 they they PRON bracis-19045 46 38 all all PRON bracis-19045 46 39 belong belong VERB bracis-19045 46 40 to to ADP bracis-19045 46 41 an an DET bracis-19045 46 42 euclidean euclidean ADJ bracis-19045 46 43 subspace subspace NOUN bracis-19045 47 1 [ [ X bracis-19045 47 2 9 9 NUM bracis-19045 47 3 ] ] SYM bracis-19045 47 4 ) ) PUNCT bracis-19045 47 5 . . PUNCT bracis-19045 48 1 hence hence ADV bracis-19045 48 2 , , PUNCT bracis-19045 48 3 it it PRON bracis-19045 48 4 is be AUX bracis-19045 48 5 possible possible ADJ bracis-19045 48 6 to to PART bracis-19045 48 7 characterize characterize VERB bracis-19045 48 8 the the DET bracis-19045 48 9 local local ADJ bracis-19045 48 10 geometry geometry NOUN bracis-19045 48 11 by by ADP bracis-19045 48 12 linear linear PROPN bracis-19045 48 13 coefficients coefficient NOUN bracis-19045 48 14 : : PUNCT bracis-19045 48 15 $ $ SYM bracis-19045 48 16 $ $ SYM bracis-19045 48 17 \begin{aligned \begin{aligne VERB bracis-19045 48 18 } } PUNCT bracis-19045 48 19 \hat{\vec \hat{\vec NOUN bracis-19045 48 20 { { PUNCT bracis-19045 48 21 x}}_i x}}_i PROPN bracis-19045 48 22 \approx \approx PROPN bracis-19045 48 23 \sum \sum NOUN bracis-19045 48 24 _ _ PUNCT bracis-19045 48 25 j j PROPN bracis-19045 48 26 w_{ij w_{ij PROPN bracis-19045 48 27 } } PUNCT bracis-19045 48 28 \vec \vec PROPN bracis-19045 48 29 { { PUNCT bracis-19045 48 30 x}_j x}_j PROPN bracis-19045 48 31 \qquad \qquad PROPN bracis-19045 48 32 \text \text PROPN bracis-19045 48 33 { { PUNCT bracis-19045 48 34 for for ADP bracis-19045 48 35 } } PUNCT bracis-19045 48 36 \qquad \qquad PROPN bracis-19045 48 37 \vec \vec PROPN bracis-19045 48 38 { { PUNCT bracis-19045 48 39 x}_j x}_j PROPN bracis-19045 48 40 \in \in PROPN bracis-19045 48 41 n(\vec n(\vec NUM bracis-19045 48 42 { { PUNCT bracis-19045 48 43 x}_i x}_i NUM bracis-19045 48 44 ) ) PUNCT bracis-19045 48 45 \end{aligned}$$ \end{aligned}$$ X bracis-19045 49 1 ( ( PUNCT bracis-19045 49 2 4 4 NUM bracis-19045 49 3 ) ) PUNCT bracis-19045 49 4 consequently consequently ADV bracis-19045 49 5 , , PUNCT bracis-19045 49 6 we we PRON bracis-19045 49 7 can can AUX bracis-19045 49 8 reconstruct reconstruct VERB bracis-19045 49 9 a a DET bracis-19045 49 10 vector vector NOUN bracis-19045 49 11 as as ADP bracis-19045 49 12 a a DET bracis-19045 49 13 linear linear ADJ bracis-19045 49 14 combination combination NOUN bracis-19045 49 15 of of ADP bracis-19045 49 16 its its PRON bracis-19045 49 17 neighbors neighbor NOUN bracis-19045 49 18 . . PUNCT bracis-19045 50 1 the the DET bracis-19045 50 2 lle lle PROPN bracis-19045 50 3 algorithm algorithm NOUN bracis-19045 50 4 requires require VERB bracis-19045 50 5 as as ADP bracis-19045 50 6 input input NOUN bracis-19045 50 7 an an DET bracis-19045 50 8 \(n \(n NOUN bracis-19045 50 9 \times \times ADP bracis-19045 50 10 m\ m\ NOUN bracis-19045 50 11 ) ) PUNCT bracis-19045 50 12 data datum NOUN bracis-19045 50 13 matrix matrix NOUN bracis-19045 50 14 x x NOUN bracis-19045 50 15 , , PUNCT bracis-19045 50 16 with with ADP bracis-19045 50 17 rows row NOUN bracis-19045 50 18 \(\vec \(\vec X bracis-19045 50 19 { { PUNCT bracis-19045 50 20 x}_i\ x}_i\ NOUN bracis-19045 50 21 ) ) PUNCT bracis-19045 50 22 , , PUNCT bracis-19045 50 23 a a DET bracis-19045 50 24 defined define VERB bracis-19045 50 25 number number NOUN bracis-19045 50 26 of of ADP bracis-19045 50 27 dimensions dimension NOUN bracis-19045 50 28 \(d \(d NOUN bracis-19045 50 29 < < X bracis-19045 50 30 m\ m\ NOUN bracis-19045 50 31 ) ) PUNCT bracis-19045 50 32 , , PUNCT bracis-19045 50 33 and and CCONJ bracis-19045 50 34 an an DET bracis-19045 50 35 integer integer NOUN bracis-19045 50 36 \(k \(k NOUN bracis-19045 50 37 > > PUNCT bracis-19045 51 1 d d PROPN bracis-19045 52 1 + + PROPN bracis-19045 52 2 1\ 1\ NUM bracis-19045 52 3 ) ) PUNCT bracis-19045 52 4 for for ADP bracis-19045 52 5 finding find VERB bracis-19045 52 6 local local ADJ bracis-19045 52 7 neighborhoods neighborhood NOUN bracis-19045 52 8 . . PUNCT bracis-19045 53 1 the the DET bracis-19045 53 2 output output NOUN bracis-19045 53 3 is be AUX bracis-19045 53 4 an an DET bracis-19045 53 5 \(n \(n NOUN bracis-19045 53 6 \times \times ADP bracis-19045 53 7 d\ d\ NOUN bracis-19045 53 8 ) ) PUNCT bracis-19045 53 9 matrix matrix NOUN bracis-19045 53 10 y y PROPN bracis-19045 53 11 , , PUNCT bracis-19045 53 12 with with ADP bracis-19045 53 13 rows row NOUN bracis-19045 53 14 \(\vec \(\vec X bracis-19045 53 15 { { PUNCT bracis-19045 53 16 y}_i\ y}_i\ NOUN bracis-19045 53 17 ) ) PUNCT bracis-19045 53 18 . . PUNCT bracis-19045 54 1 the the DET bracis-19045 54 2 lle lle PROPN bracis-19045 54 3 algorithm algorithm NOUN bracis-19045 54 4 may may AUX bracis-19045 54 5 be be AUX bracis-19045 54 6 divided divide VERB bracis-19045 54 7 into into ADP bracis-19045 54 8 three three NUM bracis-19045 54 9 main main ADJ bracis-19045 54 10 steps step NOUN bracis-19045 54 11 [ [ X bracis-19045 54 12 9 9 NUM bracis-19045 54 13 , , PUNCT bracis-19045 54 14 10 10 NUM bracis-19045 54 15 ] ] PUNCT bracis-19045 54 16 , , PUNCT bracis-19045 54 17 as as SCONJ bracis-19045 54 18 follows follow VERB bracis-19045 54 19 : : PUNCT bracis-19045 54 20 1 1 X bracis-19045 54 21 . . X bracis-19045 54 22 from from ADP bracis-19045 54 23 each each DET bracis-19045 54 24 \(\vec \(\vec NOUN bracis-19045 54 25 { { PUNCT bracis-19045 54 26 x}_i x}_i PROPN bracis-19045 54 27 \in \in PROPN bracis-19045 54 28 r^m\ r^m\ PUNCT bracis-19045 54 29 ) ) PUNCT bracis-19045 54 30 find find VERB bracis-19045 54 31 its its PRON bracis-19045 54 32 k k PROPN bracis-19045 54 33 nearest near ADJ bracis-19045 54 34 neighbors neighbor NOUN bracis-19045 54 35 ; ; PUNCT bracis-19045 54 36 2 2 X bracis-19045 54 37 . . X bracis-19045 54 38 find find VERB bracis-19045 54 39 the the DET bracis-19045 54 40 weight weight NOUN bracis-19045 54 41 matrix matrix NOUN bracis-19045 54 42 w w NOUN bracis-19045 54 43 that that PRON bracis-19045 54 44 minimizes minimize VERB bracis-19045 54 45 the the DET bracis-19045 54 46 reconstruction reconstruction NOUN bracis-19045 54 47 error error NOUN bracis-19045 54 48 for for ADP bracis-19045 54 49 each each DET bracis-19045 54 50 data data NOUN bracis-19045 54 51 point point NOUN bracis-19045 54 52 \(\vec \(\vec PUNCT bracis-19045 54 53 { { PUNCT bracis-19045 54 54 x}_i x}_i PROPN bracis-19045 54 55 \in \in PROPN bracis-19045 54 56 r^m\ r^m\ NOUN bracis-19045 54 57 ) ) PUNCT bracis-19045 54 58 ; ; PUNCT bracis-19045 54 59 3 3 X bracis-19045 54 60 . . X bracis-19045 54 61 find find VERB bracis-19045 54 62 the the DET bracis-19045 54 63 coordinates coordinate NOUN bracis-19045 54 64 y y PRON bracis-19045 54 65 which which PRON bracis-19045 54 66 minimize minimize VERB bracis-19045 54 67 the the DET bracis-19045 54 68 reconstruction reconstruction NOUN bracis-19045 54 69 error error NOUN bracis-19045 54 70 using use VERB bracis-19045 54 71 the the DET bracis-19045 54 72 optimum optimum ADJ bracis-19045 54 73 weights weight NOUN bracis-19045 54 74 . . PUNCT bracis-19045 55 1 in in ADP bracis-19045 55 2 the the DET bracis-19045 55 3 following follow VERB bracis-19045 55 4 subsections subsection NOUN bracis-19045 55 5 , , PUNCT bracis-19045 55 6 we we PRON bracis-19045 55 7 describe describe VERB bracis-19045 55 8 how how SCONJ bracis-19045 55 9 to to PART bracis-19045 55 10 obtain obtain VERB bracis-19045 55 11 the the DET bracis-19045 55 12 solution solution NOUN bracis-19045 55 13 to to ADP bracis-19045 55 14 each each DET bracis-19045 55 15 step step NOUN bracis-19045 55 16 of of ADP bracis-19045 55 17 the the DET bracis-19045 55 18 lle lle PROPN bracis-19045 55 19 algorithm algorithm NOUN bracis-19045 55 20 . . PUNCT bracis-19045 56 1 finding find VERB bracis-19045 56 2 local local ADJ bracis-19045 56 3 linear linear ADJ bracis-19045 56 4 neighborhoods neighborhood NOUN bracis-19045 56 5 . . PUNCT bracis-19045 57 1 a a DET bracis-19045 57 2 relevant relevant ADJ bracis-19045 57 3 aspect aspect NOUN bracis-19045 57 4 of of ADP bracis-19045 57 5 the the DET bracis-19045 57 6 lle lle NOUN bracis-19045 57 7 is be AUX bracis-19045 57 8 that that SCONJ bracis-19045 57 9 this this DET bracis-19045 57 10 algorithm algorithm NOUN bracis-19045 57 11 is be AUX bracis-19045 57 12 capable capable ADJ bracis-19045 57 13 of of ADP bracis-19045 57 14 recovering recover VERB bracis-19045 57 15 embeddings embedding NOUN bracis-19045 57 16 which which PRON bracis-19045 57 17 intrinsic intrinsic ADJ bracis-19045 57 18 dimensionality dimensionality NOUN bracis-19045 57 19 d d NOUN bracis-19045 57 20 is be AUX bracis-19045 57 21 smaller small ADJ bracis-19045 57 22 than than ADP bracis-19045 57 23 the the DET bracis-19045 57 24 number number NOUN bracis-19045 57 25 of of ADP bracis-19045 57 26 neighbors neighbor NOUN bracis-19045 57 27 , , PUNCT bracis-19045 57 28 k. k. PROPN bracis-19045 58 1 moreover moreover ADV bracis-19045 58 2 , , PUNCT bracis-19045 58 3 the the DET bracis-19045 58 4 assumption assumption NOUN bracis-19045 58 5 of of ADP bracis-19045 58 6 a a DET bracis-19045 58 7 linear linear ADJ bracis-19045 58 8 patch patch NOUN bracis-19045 58 9 imposes impose VERB bracis-19045 58 10 the the DET bracis-19045 58 11 existence existence NOUN bracis-19045 58 12 of of ADP bracis-19045 58 13 an an DET bracis-19045 58 14 upper upper ADJ bracis-19045 58 15 bound bind VERB bracis-19045 58 16 on on ADP bracis-19045 58 17 k. k. PROPN bracis-19045 58 18 for for ADP bracis-19045 58 19 instance instance NOUN bracis-19045 58 20 , , PUNCT bracis-19045 58 21 in in ADP bracis-19045 58 22 highly highly ADV bracis-19045 58 23 curved curved ADJ bracis-19045 58 24 datasets dataset NOUN bracis-19045 58 25 , , PUNCT bracis-19045 58 26 it it PRON bracis-19045 58 27 is be AUX bracis-19045 58 28 not not PART bracis-19045 58 29 reasonable reasonable ADJ bracis-19045 58 30 to to PART bracis-19045 58 31 have have VERB bracis-19045 58 32 a a DET bracis-19045 58 33 large large ADJ bracis-19045 58 34 k k NOUN bracis-19045 58 35 , , PUNCT bracis-19045 58 36 otherwise otherwise ADV bracis-19045 58 37 such such DET bracis-19045 58 38 an an DET bracis-19045 58 39 assumption assumption NOUN bracis-19045 58 40 would would AUX bracis-19045 58 41 be be AUX bracis-19045 58 42 violated violate VERB bracis-19045 58 43 . . PUNCT bracis-19045 59 1 in in ADP bracis-19045 59 2 the the DET bracis-19045 59 3 uncommon uncommon ADJ bracis-19045 59 4 situation situation NOUN bracis-19045 59 5 where where SCONJ bracis-19045 59 6 \(k \(k NOUN bracis-19045 59 7 > > X bracis-19045 59 8 m\ m\ NOUN bracis-19045 59 9 ) ) PUNCT bracis-19045 59 10 , , PUNCT bracis-19045 59 11 it it PRON bracis-19045 59 12 has have AUX bracis-19045 59 13 been be AUX bracis-19045 59 14 shown show VERB bracis-19045 59 15 that that SCONJ bracis-19045 59 16 each each DET bracis-19045 59 17 sample sample NOUN bracis-19045 59 18 may may AUX bracis-19045 59 19 be be AUX bracis-19045 59 20 perfectly perfectly ADV bracis-19045 59 21 be be AUX bracis-19045 59 22 reconstructed reconstruct VERB bracis-19045 59 23 from from ADP bracis-19045 59 24 its its PRON bracis-19045 59 25 neighbors neighbor NOUN bracis-19045 59 26 , , PUNCT bracis-19045 59 27 from from ADP bracis-19045 59 28 which which PRON bracis-19045 59 29 another another DET bracis-19045 59 30 problem problem NOUN bracis-19045 59 31 emerges emerge VERB bracis-19045 59 32 : : PUNCT bracis-19045 59 33 the the DET bracis-19045 59 34 reconstruction reconstruction NOUN bracis-19045 59 35 weights weight NOUN bracis-19045 59 36 are be AUX bracis-19045 59 37 not not PART bracis-19045 59 38 anymore anymore ADV bracis-19045 59 39 unique unique ADJ bracis-19045 59 40 . . PUNCT bracis-19045 60 1 to to PART bracis-19045 60 2 overcome overcome VERB bracis-19045 60 3 this this DET bracis-19045 60 4 limitation limitation NOUN bracis-19045 60 5 , , PUNCT bracis-19045 60 6 some some DET bracis-19045 60 7 regularization regularization NOUN bracis-19045 60 8 is be AUX bracis-19045 60 9 necessary necessary ADJ bracis-19045 60 10 in in ADP bracis-19045 60 11 order order NOUN bracis-19045 60 12 to to PART bracis-19045 60 13 break break VERB bracis-19045 60 14 the the DET bracis-19045 60 15 degeneracy degeneracy NOUN bracis-19045 60 16 [ [ X bracis-19045 60 17 10 10 NUM bracis-19045 60 18 ] ] PUNCT bracis-19045 60 19 . . PUNCT bracis-19045 61 1 lastly lastly ADV bracis-19045 61 2 , , PUNCT bracis-19045 61 3 a a DET bracis-19045 61 4 further further ADJ bracis-19045 61 5 concern concern NOUN bracis-19045 61 6 in in ADP bracis-19045 61 7 the the DET bracis-19045 61 8 lle lle PROPN bracis-19045 61 9 algorithm algorithm NOUN bracis-19045 61 10 refers refer VERB bracis-19045 61 11 to to ADP bracis-19045 61 12 the the DET bracis-19045 61 13 connectivity connectivity NOUN bracis-19045 61 14 of of ADP bracis-19045 61 15 the the DET bracis-19045 61 16 knn knn NOUN bracis-19045 61 17 graph graph NOUN bracis-19045 61 18 . . PUNCT bracis-19045 62 1 in in ADP bracis-19045 62 2 the the DET bracis-19045 62 3 case case NOUN bracis-19045 62 4 the the DET bracis-19045 62 5 graph graph NOUN bracis-19045 62 6 contains contain VERB bracis-19045 62 7 multiple multiple ADJ bracis-19045 62 8 connected connected ADJ bracis-19045 62 9 components component NOUN bracis-19045 62 10 , , PUNCT bracis-19045 62 11 then then ADV bracis-19045 62 12 the the DET bracis-19045 62 13 lle lle NOUN bracis-19045 62 14 should should AUX bracis-19045 62 15 be be AUX bracis-19045 62 16 applied apply VERB bracis-19045 62 17 separately separately ADV bracis-19045 62 18 on on ADP bracis-19045 62 19 each each DET bracis-19045 62 20 one one NUM bracis-19045 62 21 of of ADP bracis-19045 62 22 them they PRON bracis-19045 62 23 , , PUNCT bracis-19045 62 24 otherwise otherwise ADV bracis-19045 62 25 the the DET bracis-19045 62 26 neighborhood neighborhood NOUN bracis-19045 62 27 selection selection NOUN bracis-19045 62 28 process process NOUN bracis-19045 62 29 should should AUX bracis-19045 62 30 be be AUX bracis-19045 62 31 modified modify VERB bracis-19045 62 32 to to PART bracis-19045 62 33 assure assure VERB bracis-19045 62 34 global global ADJ bracis-19045 62 35 connectivity connectivity NOUN bracis-19045 62 36 [ [ X bracis-19045 62 37 10 10 NUM bracis-19045 62 38 ] ] PUNCT bracis-19045 62 39 . . PUNCT bracis-19045 63 1 least least ADJ bracis-19045 63 2 - - PUNCT bracis-19045 63 3 squares square NOUN bracis-19045 63 4 estimation estimation NOUN bracis-19045 63 5 of of ADP bracis-19045 63 6 the the DET bracis-19045 63 7 weights weight NOUN bracis-19045 63 8 . . PUNCT bracis-19045 64 1 the the DET bracis-19045 64 2 second second ADJ bracis-19045 64 3 step step NOUN bracis-19045 64 4 of of ADP bracis-19045 64 5 the the DET bracis-19045 64 6 lle lle NOUN bracis-19045 64 7 is be AUX bracis-19045 64 8 to to PART bracis-19045 64 9 reconstruct reconstruct VERB bracis-19045 64 10 each each DET bracis-19045 64 11 data data NOUN bracis-19045 64 12 point point NOUN bracis-19045 64 13 from from ADP bracis-19045 64 14 its its PRON bracis-19045 64 15 nearest near ADJ bracis-19045 64 16 neighbors neighbor NOUN bracis-19045 64 17 . . PUNCT bracis-19045 65 1 the the DET bracis-19045 65 2 optimal optimal ADJ bracis-19045 65 3 reconstruction reconstruction NOUN bracis-19045 65 4 weights weight NOUN bracis-19045 65 5 may may AUX bracis-19045 65 6 be be AUX bracis-19045 65 7 computed compute VERB bracis-19045 65 8 in in ADP bracis-19045 65 9 closed closed ADJ bracis-19045 65 10 form form NOUN bracis-19045 65 11 . . PUNCT bracis-19045 66 1 without without ADP bracis-19045 66 2 loss loss NOUN bracis-19045 66 3 of of ADP bracis-19045 66 4 generality generality NOUN bracis-19045 66 5 , , PUNCT bracis-19045 66 6 we we PRON bracis-19045 66 7 may may AUX bracis-19045 66 8 express express VERB bracis-19045 66 9 the the DET bracis-19045 66 10 local local ADJ bracis-19045 66 11 reconstruction reconstruction NOUN bracis-19045 66 12 error error NOUN bracis-19045 66 13 at at ADP bracis-19045 66 14 point point NOUN bracis-19045 66 15 \(\vec \(\vec PUNCT bracis-19045 66 16 { { PUNCT bracis-19045 66 17 x}_i\ x}_i\ NOUN bracis-19045 66 18 ) ) PUNCT bracis-19045 66 19 as as SCONJ bracis-19045 66 20 follows follow VERB bracis-19045 66 21 : : PUNCT bracis-19045 66 22 $ $ SYM bracis-19045 66 23 $ $ SYM bracis-19045 66 24 \begin{aligned \begin{aligne VERB bracis-19045 66 25 } } PUNCT bracis-19045 66 26 e(\vec e(\vec NOUN bracis-19045 66 27 { { PUNCT bracis-19045 66 28 w w NOUN bracis-19045 66 29 } } PUNCT bracis-19045 66 30 ) ) PUNCT bracis-19045 67 1 = = SYM bracis-19045 67 2 \left\vert \left\vert NOUN bracis-19045 67 3 \sum \sum NOUN bracis-19045 67 4 _ _ PUNCT bracis-19045 67 5 { { PUNCT bracis-19045 67 6 j j NOUN bracis-19045 67 7 } } PUNCT bracis-19045 67 8 w_j(\vec w_j(\vec X bracis-19045 67 9 { { PUNCT bracis-19045 67 10 x}_i x}_i PROPN bracis-19045 67 11 \vec \vec PROPN bracis-19045 67 12 { { PUNCT bracis-19045 67 13 x}_j x}_j PROPN bracis-19045 67 14 ) ) PUNCT bracis-19045 67 15 \right\vert \right\vert NOUN bracis-19045 67 16 ^2 ^2 PUNCT bracis-19045 67 17 = = SYM bracis-19045 67 18 \sum \sum NOUN bracis-19045 67 19 _ _ PUNCT bracis-19045 68 1 j j PROPN bracis-19045 68 2 \sum \sum NOUN bracis-19045 68 3 _ _ PUNCT bracis-19045 69 1 k k PROPN bracis-19045 69 2 w_j w_j PUNCT bracis-19045 69 3 w_k w_k PUNCT bracis-19045 69 4 ( ( PUNCT bracis-19045 69 5 \vec \vec PROPN bracis-19045 69 6 { { PUNCT bracis-19045 69 7 x}_i x}_i PROPN bracis-19045 69 8 \vec \vec PROPN bracis-19045 69 9 { { PUNCT bracis-19045 69 10 x}_j)^t(\vec x}_j)^t(\vec PROPN bracis-19045 69 11 { { PUNCT bracis-19045 69 12 x}_i x}_i PROPN bracis-19045 69 13 \vec \vec PROPN bracis-19045 69 14 { { PUNCT bracis-19045 69 15 x}_k x}_k PROPN bracis-19045 69 16 ) ) PUNCT bracis-19045 69 17 \end{aligned}$$ \end{aligned}$$ X bracis-19045 69 18 ( ( PUNCT bracis-19045 69 19 5 5 X bracis-19045 69 20 ) ) PUNCT bracis-19045 69 21 defining define VERB bracis-19045 69 22 the the DET bracis-19045 69 23 local local ADJ bracis-19045 69 24 covariance covariance NOUN bracis-19045 69 25 matrix matrix NOUN bracis-19045 69 26 c c NOUN bracis-19045 69 27 as as ADP bracis-19045 69 28 : : PUNCT bracis-19045 69 29 $ $ SYM bracis-19045 69 30 $ $ SYM bracis-19045 69 31 \begin{aligned \begin{aligne VERB bracis-19045 69 32 } } PUNCT bracis-19045 69 33 c_{jk c_{jk NOUN bracis-19045 69 34 } } PUNCT bracis-19045 69 35 = = SYM bracis-19045 69 36 ( ( PUNCT bracis-19045 69 37 \vec \vec PROPN bracis-19045 69 38 { { PUNCT bracis-19045 69 39 x}_i x}_i PROPN bracis-19045 69 40 \vec \vec PROPN bracis-19045 69 41 { { PUNCT bracis-19045 69 42 x}_j)^t x}_j)^t PROPN bracis-19045 69 43 ( ( PUNCT bracis-19045 69 44 \vec \vec PROPN bracis-19045 69 45 { { PUNCT bracis-19045 69 46 x}_i x}_i PROPN bracis-19045 69 47 \vec \vec PROPN bracis-19045 69 48 { { PUNCT bracis-19045 69 49 x}_k x}_k PROPN bracis-19045 69 50 ) ) PUNCT bracis-19045 69 51 \end{aligned}$$ \end{aligned}$$ X bracis-19045 70 1 ( ( PUNCT bracis-19045 70 2 6 6 NUM bracis-19045 70 3 ) ) PUNCT bracis-19045 70 4 we we PRON bracis-19045 70 5 then then ADV bracis-19045 70 6 have have VERB bracis-19045 70 7 the the DET bracis-19045 70 8 following follow VERB bracis-19045 70 9 expression expression NOUN bracis-19045 70 10 for for ADP bracis-19045 70 11 the the DET bracis-19045 70 12 local local ADJ bracis-19045 70 13 reconstruction reconstruction NOUN bracis-19045 70 14 error error NOUN bracis-19045 70 15 : : PUNCT bracis-19045 70 16 $ $ SYM bracis-19045 70 17 $ $ SYM bracis-19045 70 18 \begin{aligned \begin{aligne VERB bracis-19045 70 19 } } PUNCT bracis-19045 70 20 e(\vec e(\vec NOUN bracis-19045 70 21 { { PUNCT bracis-19045 70 22 w w NOUN bracis-19045 70 23 } } PUNCT bracis-19045 70 24 ) ) PUNCT bracis-19045 71 1 = = SYM bracis-19045 71 2 \sum \sum NOUN bracis-19045 71 3 _ _ PUNCT bracis-19045 72 1 j j PROPN bracis-19045 72 2 \sum \sum NOUN bracis-19045 72 3 _ _ PUNCT bracis-19045 73 1 k k PROPN bracis-19045 73 2 w_j w_j PUNCT bracis-19045 73 3 c_{jk c_{jk NOUN bracis-19045 73 4 } } PUNCT bracis-19045 73 5 w_k w_k PUNCT bracis-19045 73 6 = = SYM bracis-19045 73 7 \vec \vec PROPN bracis-19045 73 8 { { PUNCT bracis-19045 73 9 w}^t w}^t PROPN bracis-19045 73 10 c c PROPN bracis-19045 73 11 \vec \vec PROPN bracis-19045 73 12 { { PUNCT bracis-19045 73 13 w w PROPN bracis-19045 73 14 } } PUNCT bracis-19045 73 15 \end{aligned}$$ \end{aligned}$$ X bracis-19045 73 16 ( ( PUNCT bracis-19045 73 17 7 7 NUM bracis-19045 73 18 ) ) PUNCT bracis-19045 73 19 regarding regard VERB bracis-19045 73 20 the the DET bracis-19045 73 21 constraint constraint NOUN bracis-19045 73 22 \(\sum \(\sum PUNCT bracis-19045 74 1 _ _ PRON bracis-19045 74 2 j j NOUN bracis-19045 74 3 w_j w_j PUNCT bracis-19045 74 4 = = SYM bracis-19045 74 5 1\ 1\ NUM bracis-19045 74 6 ) ) PUNCT bracis-19045 74 7 , , PUNCT bracis-19045 74 8 that that PRON bracis-19045 74 9 may may AUX bracis-19045 74 10 be be AUX bracis-19045 74 11 interpreted interpret VERB bracis-19045 74 12 in in ADP bracis-19045 74 13 two two NUM bracis-19045 74 14 different different ADJ bracis-19045 74 15 manners manner NOUN bracis-19045 74 16 , , PUNCT bracis-19045 74 17 namely namely ADV bracis-19045 74 18 geometrically geometrically ADV bracis-19045 74 19 and and CCONJ bracis-19045 74 20 probabilistically probabilistically ADV bracis-19045 74 21 . . PUNCT bracis-19045 75 1 from from ADP bracis-19045 75 2 a a DET bracis-19045 75 3 geometric geometric ADJ bracis-19045 75 4 point point NOUN bracis-19045 75 5 of of ADP bracis-19045 75 6 view view NOUN bracis-19045 75 7 , , PUNCT bracis-19045 75 8 it it PRON bracis-19045 75 9 provides provide VERB bracis-19045 75 10 invariance invariance NOUN bracis-19045 75 11 under under ADP bracis-19045 75 12 translation translation NOUN bracis-19045 75 13 , , PUNCT bracis-19045 75 14 thus thus ADV bracis-19045 75 15 , , PUNCT bracis-19045 75 16 by by ADP bracis-19045 75 17 adding add VERB bracis-19045 75 18 a a DET bracis-19045 75 19 given give VERB bracis-19045 75 20 constant constant ADJ bracis-19045 75 21 vector vector NOUN bracis-19045 75 22 \(\vec \(\vec NOUN bracis-19045 75 23 { { PUNCT bracis-19045 75 24 c}\ c}\ NUM bracis-19045 75 25 ) ) PUNCT bracis-19045 75 26 to to PART bracis-19045 75 27 \(\vec \(\vec NOUN bracis-19045 75 28 { { PUNCT bracis-19045 75 29 x}_i\ x}_i\ NOUN bracis-19045 75 30 ) ) PUNCT bracis-19045 75 31 and and CCONJ bracis-19045 75 32 all all PRON bracis-19045 75 33 of of ADP bracis-19045 75 34 its its PRON bracis-19045 75 35 neighbors neighbor NOUN bracis-19045 75 36 , , PUNCT bracis-19045 75 37 the the DET bracis-19045 75 38 reconstruction reconstruction NOUN bracis-19045 75 39 error error NOUN bracis-19045 75 40 remains remain VERB bracis-19045 75 41 unchanged unchanged ADJ bracis-19045 75 42 . . PUNCT bracis-19045 76 1 in in ADP bracis-19045 76 2 terms term NOUN bracis-19045 76 3 of of ADP bracis-19045 76 4 probability probability NOUN bracis-19045 76 5 , , PUNCT bracis-19045 76 6 enforcing enforce VERB bracis-19045 76 7 the the DET bracis-19045 76 8 weights weight NOUN bracis-19045 76 9 to to PART bracis-19045 76 10 sum sum VERB bracis-19045 76 11 to to ADP bracis-19045 76 12 one one NUM bracis-19045 76 13 leads lead VERB bracis-19045 76 14 w w NOUN bracis-19045 76 15 to to PART bracis-19045 76 16 become become VERB bracis-19045 76 17 a a DET bracis-19045 76 18 stochastic stochastic ADJ bracis-19045 76 19 transition transition NOUN bracis-19045 76 20 matrix matrix NOUN bracis-19045 76 21 [ [ X bracis-19045 76 22 10 10 NUM bracis-19045 76 23 ] ] PUNCT bracis-19045 76 24 directly directly ADV bracis-19045 76 25 related relate VERB bracis-19045 76 26 to to ADP bracis-19045 76 27 markov markov NOUN bracis-19045 76 28 chains chain NOUN bracis-19045 76 29 and and CCONJ bracis-19045 76 30 diffusion diffusion NOUN bracis-19045 76 31 maps map NOUN bracis-19045 76 32 [ [ X bracis-19045 76 33 11 11 NUM bracis-19045 76 34 ] ] PUNCT bracis-19045 76 35 . . PUNCT bracis-19045 77 1 as as SCONJ bracis-19045 77 2 detailed detailed ADJ bracis-19045 77 3 below below ADV bracis-19045 77 4 , , PUNCT bracis-19045 77 5 in in ADP bracis-19045 77 6 the the DET bracis-19045 77 7 minimization minimization NOUN bracis-19045 77 8 of of ADP bracis-19045 77 9 the the DET bracis-19045 77 10 squared square VERB bracis-19045 77 11 error error NOUN bracis-19045 77 12 the the DET bracis-19045 77 13 solution solution NOUN bracis-19045 77 14 is be AUX bracis-19045 77 15 found find VERB bracis-19045 77 16 through through ADP bracis-19045 77 17 an an DET bracis-19045 77 18 eigenvalue eigenvalue NOUN bracis-19045 77 19 problem problem NOUN bracis-19045 77 20 . . PUNCT bracis-19045 78 1 in in ADP bracis-19045 78 2 fact fact NOUN bracis-19045 78 3 , , PUNCT bracis-19045 78 4 the the DET bracis-19045 78 5 estimation estimation NOUN bracis-19045 78 6 of of ADP bracis-19045 78 7 the the DET bracis-19045 78 8 matrix matrix NOUN bracis-19045 78 9 w w NOUN bracis-19045 78 10 reduces reduce VERB bracis-19045 78 11 to to ADP bracis-19045 78 12 n n NUM bracis-19045 78 13 eigenvalue eigenvalue VERB bracis-19045 78 14 problems problem NOUN bracis-19045 78 15 . . PUNCT bracis-19045 79 1 considering consider VERB bracis-19045 79 2 that that SCONJ bracis-19045 79 3 there there PRON bracis-19045 79 4 are be VERB bracis-19045 79 5 no no DET bracis-19045 79 6 constraints constraint NOUN bracis-19045 79 7 across across ADP bracis-19045 79 8 the the DET bracis-19045 79 9 rows row NOUN bracis-19045 79 10 of of ADP bracis-19045 79 11 w w NOUN bracis-19045 79 12 , , PUNCT bracis-19045 79 13 we we PRON bracis-19045 79 14 may may AUX bracis-19045 79 15 then then ADV bracis-19045 79 16 find find VERB bracis-19045 79 17 the the DET bracis-19045 79 18 optimal optimal ADJ bracis-19045 79 19 weights weight NOUN bracis-19045 79 20 for for ADP bracis-19045 79 21 each each DET bracis-19045 79 22 sample sample NOUN bracis-19045 79 23 \(\vec \(\vec X bracis-19045 79 24 { { PUNCT bracis-19045 79 25 x}_i\ x}_i\ NOUN bracis-19045 79 26 ) ) PUNCT bracis-19045 79 27 separately separately ADV bracis-19045 79 28 , , PUNCT bracis-19045 79 29 drastically drastically ADV bracis-19045 79 30 simplifying simplify VERB bracis-19045 79 31 the the DET bracis-19045 79 32 respective respective ADJ bracis-19045 79 33 computations computation NOUN bracis-19045 79 34 . . PUNCT bracis-19045 80 1 therefore therefore ADV bracis-19045 80 2 , , PUNCT bracis-19045 80 3 we we PRON bracis-19045 80 4 have have VERB bracis-19045 80 5 n n ADV bracis-19045 80 6 independent independent ADJ bracis-19045 80 7 constrained constrain VERB bracis-19045 80 8 optimization optimization NOUN bracis-19045 80 9 problems problem NOUN bracis-19045 80 10 given give VERB bracis-19045 80 11 by by ADP bracis-19045 80 12 : : PUNCT bracis-19045 80 13 $ $ SYM bracis-19045 80 14 $ $ SYM bracis-19045 80 15 \begin{aligned \begin{aligne VERB bracis-19045 80 16 } } PUNCT bracis-19045 80 17 \mathop \mathop PROPN bracis-19045 80 18 { { PUNCT bracis-19045 80 19 \mathrm \mathrm PROPN bracis-19045 80 20 { { PUNCT bracis-19045 80 21 arg\,min}}\limits arg\,min}}\limits X bracis-19045 80 22 _ _ X bracis-19045 80 23 { { PUNCT bracis-19045 80 24 \vec \vec NOUN bracis-19045 80 25 { { PUNCT bracis-19045 80 26 w}_i}~ w}_i}~ PROPN bracis-19045 80 27 \vec \vec PROPN bracis-19045 80 28 { { PUNCT bracis-19045 80 29 w}_i^t w}_i^t X bracis-19045 80 30 c_i c_i SYM bracis-19045 80 31 \vec \vec PROPN bracis-19045 80 32 { { PUNCT bracis-19045 80 33 w}_i w}_i PROPN bracis-19045 80 34 \quad \quad PROPN bracis-19045 80 35 \text \text PROPN bracis-19045 80 36 { { PUNCT bracis-19045 80 37 s.t s.t PROPN bracis-19045 80 38 . . PROPN bracis-19045 80 39 } } PUNCT bracis-19045 80 40 \quad \quad PROPN bracis-19045 80 41 \vec \vec PROPN bracis-19045 80 42 { { PUNCT bracis-19045 80 43 1}^t 1}^t PROPN bracis-19045 80 44 \vec \vec PROPN bracis-19045 80 45 { { PUNCT bracis-19045 80 46 w}_i w}_i PROPN bracis-19045 80 47 = = SYM bracis-19045 80 48 1 1 NUM bracis-19045 80 49 \end{aligned}$$ \end{aligned}$$ NOUN bracis-19045 80 50 ( ( PUNCT bracis-19045 80 51 8) 8) NUM bracis-19045 80 52 for for ADP bracis-19045 80 53 \(i \(i NOUN bracis-19045 80 54 = = SYM bracis-19045 80 55 1 1 NUM bracis-19045 80 56 , , PUNCT bracis-19045 80 57 2 2 NUM bracis-19045 80 58 , , PUNCT bracis-19045 80 59 ... ... PUNCT bracis-19045 80 60 , , PUNCT bracis-19045 80 61 n\ n\ NOUN bracis-19045 80 62 ) ) PUNCT bracis-19045 80 63 . . PUNCT bracis-19045 81 1 using use VERB bracis-19045 81 2 lagrange lagrange NOUN bracis-19045 81 3 multipliers multiplier NOUN bracis-19045 81 4 , , PUNCT bracis-19045 81 5 we we PRON bracis-19045 81 6 express express VERB bracis-19045 81 7 the the DET bracis-19045 81 8 lagrangian lagrangian ADJ bracis-19045 81 9 function function NOUN bracis-19045 81 10 as as SCONJ bracis-19045 81 11 follows follow VERB bracis-19045 81 12 : : PUNCT bracis-19045 81 13 $ $ SYM bracis-19045 81 14 $ $ SYM bracis-19045 81 15 \begin{aligned \begin{aligne VERB bracis-19045 81 16 } } PUNCT bracis-19045 81 17 l(\vec l(\vec X bracis-19045 81 18 { { PUNCT bracis-19045 81 19 w}_i w}_i PROPN bracis-19045 81 20 , , PUNCT bracis-19045 81 21 \lambda \lambda PROPN bracis-19045 81 22 ) ) PUNCT bracis-19045 82 1 = = SYM bracis-19045 82 2 \vec \vec PROPN bracis-19045 82 3 { { PUNCT bracis-19045 82 4 w}_i^t w}_i^t X bracis-19045 82 5 c_i c_i SYM bracis-19045 82 6 \vec \vec PROPN bracis-19045 82 7 { { PUNCT bracis-19045 82 8 w}_i w}_i PROPN bracis-19045 82 9 \lambda \lambda PROPN bracis-19045 82 10 ( ( PUNCT bracis-19045 82 11 \vec \vec PROPN bracis-19045 82 12 { { PUNCT bracis-19045 82 13 1}^t 1}^t PROPN bracis-19045 82 14 \vec \vec PROPN bracis-19045 82 15 { { PUNCT bracis-19045 82 16 w}_i w}_i PROPN bracis-19045 82 17 1 1 NUM bracis-19045 82 18 ) ) PUNCT bracis-19045 82 19 \end{aligned}$$ \end{aligned}$$ X bracis-19045 82 20 ( ( PUNCT bracis-19045 82 21 9 9 X bracis-19045 82 22 ) ) PUNCT bracis-19045 82 23 taking take VERB bracis-19045 82 24 the the DET bracis-19045 82 25 derivatives derivative NOUN bracis-19045 82 26 with with ADP bracis-19045 82 27 relation relation NOUN bracis-19045 82 28 to to ADP bracis-19045 82 29 \(\vec \(\vec NOUN bracis-19045 82 30 { { PUNCT bracis-19045 82 31 w}_i\ w}_i\ NOUN bracis-19045 82 32 ) ) PUNCT bracis-19045 82 33 as as SCONJ bracis-19045 82 34 follows follow VERB bracis-19045 82 35 : : PUNCT bracis-19045 82 36 $ $ SYM bracis-19045 82 37 $ $ SYM bracis-19045 82 38 \begin{aligned \begin{aligne VERB bracis-19045 82 39 } } PUNCT bracis-19045 82 40 \frac{\partial \frac{\partial NOUN bracis-19045 82 41 } } PUNCT bracis-19045 82 42 { { PUNCT bracis-19045 82 43 \partial \partial PROPN bracis-19045 82 44 \vec \vec PROPN bracis-19045 82 45 { { PUNCT bracis-19045 82 46 w}_i}l(\vec w}_i}l(\vec X bracis-19045 82 47 { { PUNCT bracis-19045 82 48 w}_i w}_i PROPN bracis-19045 82 49 , , PUNCT bracis-19045 82 50 \lambda \lambda PROPN bracis-19045 82 51 ) ) PUNCT bracis-19045 82 52 = = SYM bracis-19045 83 1 2 2 NUM bracis-19045 83 2 c_i c_i SYM bracis-19045 83 3 \vec \vec PROPN bracis-19045 83 4 { { PUNCT bracis-19045 83 5 w}_i w}_i PROPN bracis-19045 83 6 \lambda \lambda PROPN bracis-19045 83 7 \vec \vec PROPN bracis-19045 83 8 { { PUNCT bracis-19045 83 9 1 1 NUM bracis-19045 83 10 } } PUNCT bracis-19045 83 11 = = SYM bracis-19045 83 12 0 0 NUM bracis-19045 83 13 \end{aligned}$$ \end{aligned}$$ PROPN bracis-19045 83 14 ( ( PUNCT bracis-19045 83 15 10 10 NUM bracis-19045 83 16 ) ) PUNCT bracis-19045 83 17 leads lead VERB bracis-19045 83 18 to to ADP bracis-19045 83 19 $ $ SYM bracis-19045 83 20 $ $ SYM bracis-19045 83 21 \begin{aligned \begin{aligne VERB bracis-19045 83 22 } } PUNCT bracis-19045 83 23 c_i c_i SYM bracis-19045 83 24 \vec \vec PROPN bracis-19045 83 25 { { PUNCT bracis-19045 83 26 w}_i w}_i PROPN bracis-19045 83 27 = = SYM bracis-19045 83 28 \frac{\lambda \frac{\lambda NOUN bracis-19045 83 29 } } PUNCT bracis-19045 83 30 { { PUNCT bracis-19045 83 31 2}\vec 2}\vec NUM bracis-19045 83 32 { { PUNCT bracis-19045 83 33 1 1 NUM bracis-19045 83 34 } } PUNCT bracis-19045 83 35 \end{aligned}$$ \end{aligned}$$ X bracis-19045 83 36 ( ( PUNCT bracis-19045 83 37 11 11 NUM bracis-19045 83 38 ) ) PUNCT bracis-19045 83 39 which which PRON bracis-19045 83 40 is be AUX bracis-19045 83 41 equivalent equivalent ADJ bracis-19045 83 42 to to ADP bracis-19045 83 43 solving solve VERB bracis-19045 83 44 the the DET bracis-19045 83 45 following follow VERB bracis-19045 83 46 linear linear ADJ bracis-19045 83 47 system system NOUN bracis-19045 83 48 : : PUNCT bracis-19045 83 49 $ $ SYM bracis-19045 83 50 $ $ SYM bracis-19045 83 51 \begin{aligned \begin{aligne VERB bracis-19045 83 52 } } PUNCT bracis-19045 83 53 c_i c_i SYM bracis-19045 83 54 \vec \vec PROPN bracis-19045 83 55 { { PUNCT bracis-19045 83 56 w}_i w}_i PROPN bracis-19045 83 57 = = SYM bracis-19045 83 58 \vec \vec PROPN bracis-19045 83 59 { { PUNCT bracis-19045 83 60 1 1 NUM bracis-19045 83 61 } } PUNCT bracis-19045 83 62 \end{aligned}$$ \end{aligned}$$ X bracis-19045 83 63 ( ( PUNCT bracis-19045 83 64 12 12 NUM bracis-19045 83 65 ) ) PUNCT bracis-19045 83 66 and and CCONJ bracis-19045 83 67 then then ADV bracis-19045 83 68 normalizing normalize VERB bracis-19045 83 69 the the DET bracis-19045 83 70 solution solution NOUN bracis-19045 83 71 to to PART bracis-19045 83 72 guarantee guarantee VERB bracis-19045 83 73 that that DET bracis-19045 83 74 \(\sum \(\sum VERB bracis-19045 84 1 _ _ PRON bracis-19045 84 2 j j NOUN bracis-19045 84 3 w_i(j w_i(j ADV bracis-19045 84 4 ) ) PUNCT bracis-19045 84 5 = = SYM bracis-19045 84 6 1\ 1\ NUM bracis-19045 84 7 ) ) PUNCT bracis-19045 84 8 by by ADP bracis-19045 84 9 dividing divide VERB bracis-19045 84 10 each each DET bracis-19045 84 11 coefficient coefficient NOUN bracis-19045 84 12 of of ADP bracis-19045 84 13 the the DET bracis-19045 84 14 vector vector NOUN bracis-19045 84 15 \(\vec \(\vec NOUN bracis-19045 84 16 { { PUNCT bracis-19045 84 17 w}_i\ w}_i\ NOUN bracis-19045 84 18 ) ) PUNCT bracis-19045 84 19 by by ADP bracis-19045 84 20 the the DET bracis-19045 84 21 sum sum NOUN bracis-19045 84 22 of of ADP bracis-19045 84 23 all all DET bracis-19045 84 24 the the DET bracis-19045 84 25 coefficients coefficient NOUN bracis-19045 84 26 : : PUNCT bracis-19045 84 27 $ $ SYM bracis-19045 84 28 $ $ SYM bracis-19045 84 29 \begin{aligned \begin{aligne VERB bracis-19045 84 30 } } PUNCT bracis-19045 84 31 w_i(j w_i(j ADV bracis-19045 84 32 ) ) PUNCT bracis-19045 84 33 = = NOUN bracis-19045 84 34 \frac{\displaystyle \frac{\displaystyle NOUN bracis-19045 84 35 w_i(j)}{\displaystyle w_i(j)}{\displaystyle NOUN bracis-19045 84 36 \sum \sum NOUN bracis-19045 84 37 _ _ PUNCT bracis-19045 84 38 j j PROPN bracis-19045 84 39 w_i(j w_i(j ADV bracis-19045 84 40 ) ) PUNCT bracis-19045 84 41 } } PUNCT bracis-19045 85 1 \qquad \qquad PRON bracis-19045 85 2 \text \text ADV bracis-19045 85 3 { { PUNCT bracis-19045 85 4 for for ADP bracis-19045 85 5 } } PUNCT bracis-19045 85 6 \qquad \qquad PROPN bracis-19045 85 7 j=1,2, j=1,2, NOUN bracis-19045 85 8 ... ... PUNCT bracis-19045 85 9 ,m ,m PUNCT bracis-19045 85 10 \end{aligned}$$ \end{aligned}$$ X bracis-19045 85 11 ( ( PUNCT bracis-19045 85 12 13 13 NUM bracis-19045 85 13 ) ) PUNCT bracis-19045 85 14 in in ADP bracis-19045 85 15 the the DET bracis-19045 85 16 case case NOUN bracis-19045 85 17 that that SCONJ bracis-19045 85 18 k k PROPN bracis-19045 85 19 ( ( PUNCT bracis-19045 85 20 i.e. i.e. X bracis-19045 85 21 , , PUNCT bracis-19045 85 22 number number NOUN bracis-19045 85 23 of of ADP bracis-19045 85 24 neighbors neighbor NOUN bracis-19045 85 25 ) ) PUNCT bracis-19045 85 26 is be AUX bracis-19045 85 27 greater great ADJ bracis-19045 85 28 than than SCONJ bracis-19045 85 29 m m PROPN bracis-19045 85 30 ( ( PUNCT bracis-19045 85 31 i.e. i.e. X bracis-19045 85 32 , , PUNCT bracis-19045 85 33 number number NOUN bracis-19045 85 34 of of ADP bracis-19045 85 35 features feature NOUN bracis-19045 85 36 ) ) PUNCT bracis-19045 85 37 then then ADV bracis-19045 85 38 , , PUNCT bracis-19045 85 39 in in ADP bracis-19045 85 40 general general ADJ bracis-19045 85 41 , , PUNCT bracis-19045 85 42 the the DET bracis-19045 85 43 space space NOUN bracis-19045 85 44 spanned span VERB bracis-19045 85 45 by by ADP bracis-19045 85 46 k k PROPN bracis-19045 85 47 distinct distinct ADJ bracis-19045 85 48 vectors vector NOUN bracis-19045 85 49 consists consist VERB bracis-19045 85 50 of of ADP bracis-19045 85 51 the the DET bracis-19045 85 52 whole whole ADJ bracis-19045 85 53 space space NOUN bracis-19045 85 54 . . PUNCT bracis-19045 86 1 this this PRON bracis-19045 86 2 means mean VERB bracis-19045 86 3 that that SCONJ bracis-19045 86 4 \(\vec \(\vec NOUN bracis-19045 86 5 { { PUNCT bracis-19045 86 6 x}_i\ x}_i\ NOUN bracis-19045 86 7 ) ) PUNCT bracis-19045 86 8 may may AUX bracis-19045 86 9 be be AUX bracis-19045 86 10 expressed express VERB bracis-19045 86 11 exactly exactly ADV bracis-19045 86 12 as as ADP bracis-19045 86 13 a a DET bracis-19045 86 14 linear linear ADJ bracis-19045 86 15 combination combination NOUN bracis-19045 86 16 of of ADP bracis-19045 86 17 its its PRON bracis-19045 86 18 k k NOUN bracis-19045 86 19 - - PUNCT bracis-19045 86 20 nearest near ADJ bracis-19045 86 21 neighbors neighbor NOUN bracis-19045 86 22 . . PUNCT bracis-19045 87 1 in in ADP bracis-19045 87 2 fact fact NOUN bracis-19045 87 3 , , PUNCT bracis-19045 87 4 if if SCONJ bracis-19045 87 5 \(k \(k NOUN bracis-19045 87 6 > > X bracis-19045 87 7 m\ m\ NOUN bracis-19045 87 8 ) ) PUNCT bracis-19045 87 9 , , PUNCT bracis-19045 87 10 there there PRON bracis-19045 87 11 are be VERB bracis-19045 87 12 generally generally ADV bracis-19045 87 13 infinitely infinitely ADV bracis-19045 87 14 many many ADJ bracis-19045 87 15 solutions solution NOUN bracis-19045 87 16 to to ADP bracis-19045 87 17 \(\vec \(\vec NOUN bracis-19045 87 18 { { PUNCT bracis-19045 87 19 x}_i x}_i PROPN bracis-19045 87 20 = = SYM bracis-19045 87 21 \sum \sum NOUN bracis-19045 87 22 _ _ PUNCT bracis-19045 87 23 j j PROPN bracis-19045 87 24 w_j w_j SYM bracis-19045 87 25 \vec \vec PROPN bracis-19045 87 26 { { PUNCT bracis-19045 87 27 x}_j\ x}_j\ PROPN bracis-19045 87 28 ) ) PUNCT bracis-19045 87 29 , , PUNCT bracis-19045 87 30 due due ADP bracis-19045 87 31 to to ADP bracis-19045 87 32 the the DET bracis-19045 87 33 fact fact NOUN bracis-19045 87 34 that that SCONJ bracis-19045 87 35 there there PRON bracis-19045 87 36 would would AUX bracis-19045 87 37 be be AUX bracis-19045 87 38 more more ADJ bracis-19045 87 39 unknowns unknown NOUN bracis-19045 87 40 ( ( PUNCT bracis-19045 87 41 k k NOUN bracis-19045 87 42 ) ) PUNCT bracis-19045 87 43 than than ADP bracis-19045 87 44 equations equation NOUN bracis-19045 87 45 ( ( PUNCT bracis-19045 87 46 m m NOUN bracis-19045 87 47 ) ) PUNCT bracis-19045 87 48 . . PUNCT bracis-19045 88 1 in in ADP bracis-19045 88 2 such such DET bracis-19045 88 3 a a DET bracis-19045 88 4 case case NOUN bracis-19045 88 5 , , PUNCT bracis-19045 88 6 the the DET bracis-19045 88 7 optimization optimization NOUN bracis-19045 88 8 problem problem NOUN bracis-19045 88 9 is be AUX bracis-19045 88 10 ill ill ADV bracis-19045 88 11 - - PUNCT bracis-19045 88 12 posed pose VERB bracis-19045 88 13 and and CCONJ bracis-19045 88 14 , , PUNCT bracis-19045 88 15 thus thus ADV bracis-19045 88 16 , , PUNCT bracis-19045 88 17 regularization regularization NOUN bracis-19045 88 18 is be AUX bracis-19045 88 19 required require VERB bracis-19045 88 20 . . PUNCT bracis-19045 89 1 a a DET bracis-19045 89 2 common common ADJ bracis-19045 89 3 regularization regularization NOUN bracis-19045 89 4 technique technique NOUN bracis-19045 89 5 refers refer VERB bracis-19045 89 6 to to ADP bracis-19045 89 7 the the DET bracis-19045 89 8 tikonov tikonov NOUN bracis-19045 89 9 regularization regularization NOUN bracis-19045 89 10 , , PUNCT bracis-19045 89 11 which which PRON bracis-19045 89 12 adds add VERB bracis-19045 89 13 a a DET bracis-19045 89 14 penalization penalization NOUN bracis-19045 89 15 term term NOUN bracis-19045 89 16 to to ADP bracis-19045 89 17 the the DET bracis-19045 89 18 least least ADJ bracis-19045 89 19 squares square NOUN bracis-19045 89 20 problem problem NOUN bracis-19045 89 21 , , PUNCT bracis-19045 89 22 as as SCONJ bracis-19045 89 23 follows follow VERB bracis-19045 89 24 : : PUNCT bracis-19045 89 25 $ $ SYM bracis-19045 89 26 $ $ SYM bracis-19045 89 27 \begin{aligned \begin{aligne VERB bracis-19045 89 28 } } PUNCT bracis-19045 89 29 \left\vert \left\vert NOUN bracis-19045 89 30 { { PUNCT bracis-19045 89 31 \vec \vec PROPN bracis-19045 89 32 { { PUNCT bracis-19045 89 33 x}}_i x}}_i PROPN bracis-19045 89 34 \sum \sum NOUN bracis-19045 89 35 _ _ PUNCT bracis-19045 90 1 j j PROPN bracis-19045 90 2 w_j w_j PUNCT bracis-19045 90 3 { { PUNCT bracis-19045 90 4 \vec \vec PROPN bracis-19045 90 5 { { PUNCT bracis-19045 90 6 x}}_j x}}_j PROPN bracis-19045 90 7 \right\vert \right\vert PROPN bracis-19045 90 8 ^2 ^2 PUNCT bracis-19045 90 9 + + PUNCT bracis-19045 90 10 \alpha \alpha ADJ bracis-19045 90 11 \sum \sum NOUN bracis-19045 90 12 _ _ PUNCT bracis-19045 90 13 j j PROPN bracis-19045 91 1 w_j^2 w_j^2 X bracis-19045 91 2 \end{aligned}$$ \end{aligned}$$ X bracis-19045 91 3 ( ( PUNCT bracis-19045 91 4 14 14 NUM bracis-19045 91 5 ) ) PUNCT bracis-19045 91 6 where where SCONJ bracis-19045 91 7 \(\alpha \(\alpha ADV bracis-19045 91 8 \ \ NOUN bracis-19045 91 9 ) ) PUNCT bracis-19045 91 10 controls control VERB bracis-19045 91 11 the the DET bracis-19045 91 12 degree degree NOUN bracis-19045 91 13 of of ADP bracis-19045 91 14 regularization regularization NOUN bracis-19045 91 15 . . PUNCT bracis-19045 92 1 in in ADP bracis-19045 92 2 other other ADJ bracis-19045 92 3 words word NOUN bracis-19045 92 4 , , PUNCT bracis-19045 92 5 it it PRON bracis-19045 92 6 selects select VERB bracis-19045 92 7 the the DET bracis-19045 92 8 weights weight NOUN bracis-19045 92 9 which which PRON bracis-19045 92 10 minimize minimize VERB bracis-19045 92 11 a a DET bracis-19045 92 12 combination combination NOUN bracis-19045 92 13 of of ADP bracis-19045 92 14 reconstruction reconstruction NOUN bracis-19045 92 15 error error NOUN bracis-19045 92 16 as as ADV bracis-19045 92 17 well well ADV bracis-19045 92 18 as as ADP bracis-19045 92 19 the the DET bracis-19045 92 20 sum sum NOUN bracis-19045 92 21 of of ADP bracis-19045 92 22 the the DET bracis-19045 92 23 squared square VERB bracis-19045 92 24 weights weight NOUN bracis-19045 92 25 . . PUNCT bracis-19045 93 1 in in ADP bracis-19045 93 2 the the DET bracis-19045 93 3 case case NOUN bracis-19045 93 4 that that PRON bracis-19045 93 5 \(\alpha \(\alpha VERB bracis-19045 93 6 \rightarrow \rightarrow PROPN bracis-19045 93 7 0\ 0\ PROPN bracis-19045 93 8 ) ) PUNCT bracis-19045 93 9 , , PUNCT bracis-19045 93 10 then then ADV bracis-19045 93 11 there there PRON bracis-19045 93 12 is be VERB bracis-19045 93 13 a a DET bracis-19045 93 14 least least ADJ bracis-19045 93 15 - - PUNCT bracis-19045 93 16 squares square NOUN bracis-19045 93 17 problem problem NOUN bracis-19045 93 18 . . PUNCT bracis-19045 94 1 however however ADV bracis-19045 94 2 , , PUNCT bracis-19045 94 3 in in ADP bracis-19045 94 4 the the DET bracis-19045 94 5 opposite opposite ADJ bracis-19045 94 6 limit limit NOUN bracis-19045 94 7 ( ( PUNCT bracis-19045 94 8 i.e. i.e. X bracis-19045 94 9 , , PUNCT bracis-19045 94 10 \(\alpha \(\alpha ADV bracis-19045 94 11 \rightarrow \rightarrow PROPN bracis-19045 94 12 \infty \infty PROPN bracis-19045 94 13 \ \ PROPN bracis-19045 94 14 ) ) PUNCT bracis-19045 94 15 ) ) PUNCT bracis-19045 94 16 , , PUNCT bracis-19045 94 17 the the DET bracis-19045 94 18 squared square VERB bracis-19045 94 19 - - PUNCT bracis-19045 94 20 error error NOUN bracis-19045 94 21 term term NOUN bracis-19045 94 22 becomes become VERB bracis-19045 94 23 negligible negligible ADJ bracis-19045 94 24 , , PUNCT bracis-19045 94 25 allowing allow VERB bracis-19045 94 26 the the DET bracis-19045 94 27 minimization minimization NOUN bracis-19045 94 28 of of ADP bracis-19045 94 29 the the DET bracis-19045 94 30 euclidean euclidean ADJ bracis-19045 94 31 norm norm NOUN bracis-19045 94 32 of of ADP bracis-19045 94 33 the the DET bracis-19045 94 34 weight weight NOUN bracis-19045 94 35 vector vector NOUN bracis-19045 94 36 \(\vec \(\vec NOUN bracis-19045 94 37 { { PUNCT bracis-19045 94 38 w}\ w}\ PROPN bracis-19045 94 39 ) ) PUNCT bracis-19045 94 40 . . PUNCT bracis-19045 95 1 typically typically ADV bracis-19045 95 2 , , PUNCT bracis-19045 95 3 \(\alpha \(\alpha PROPN bracis-19045 95 4 \ \ NOUN bracis-19045 95 5 ) ) PUNCT bracis-19045 95 6 is be AUX bracis-19045 95 7 set set VERB bracis-19045 95 8 to to PART bracis-19045 95 9 be be AUX bracis-19045 95 10 a a DET bracis-19045 95 11 small small ADJ bracis-19045 95 12 but but CCONJ bracis-19045 95 13 non non ADJ bracis-19045 95 14 - - ADJ bracis-19045 95 15 zero zero NUM bracis-19045 95 16 value value NOUN bracis-19045 95 17 . . PUNCT bracis-19045 96 1 in in ADP bracis-19045 96 2 this this DET bracis-19045 96 3 case case NOUN bracis-19045 96 4 , , PUNCT bracis-19045 96 5 the the DET bracis-19045 96 6 n n CCONJ bracis-19045 96 7 independent independent ADJ bracis-19045 96 8 constrained constrain VERB bracis-19045 96 9 optimization optimization NOUN bracis-19045 96 10 problems problem NOUN bracis-19045 96 11 are be AUX bracis-19045 96 12 : : PUNCT bracis-19045 96 13 $ $ SYM bracis-19045 96 14 $ $ SYM bracis-19045 96 15 \begin{aligned \begin{aligne VERB bracis-19045 96 16 } } PUNCT bracis-19045 96 17 \mathop \mathop PROPN bracis-19045 96 18 { { PUNCT bracis-19045 96 19 \mathrm \mathrm PROPN bracis-19045 96 20 { { PUNCT bracis-19045 96 21 arg\,min}}\limits arg\,min}}\limits X bracis-19045 96 22 _ _ X bracis-19045 96 23 { { PUNCT bracis-19045 96 24 \vec \vec NOUN bracis-19045 96 25 { { PUNCT bracis-19045 96 26 w}_i}~ w}_i}~ PROPN bracis-19045 96 27 \vec \vec PROPN bracis-19045 96 28 { { PUNCT bracis-19045 96 29 w}_i^t w}_i^t X bracis-19045 96 30 c_i c_i SYM bracis-19045 96 31 \vec \vec NOUN bracis-19045 96 32 { { PUNCT bracis-19045 96 33 w}_i w}_i PROPN bracis-19045 96 34 + + SYM bracis-19045 96 35 \alpha \alpha PROPN bracis-19045 96 36 \vec \vec PROPN bracis-19045 96 37 { { PUNCT bracis-19045 96 38 w}_i^t w}_i^t X bracis-19045 96 39 \vec \vec PROPN bracis-19045 96 40 { { PUNCT bracis-19045 96 41 w}_i w}_i PROPN bracis-19045 96 42 \quad \quad PROPN bracis-19045 96 43 \text \text PROPN bracis-19045 96 44 { { PUNCT bracis-19045 96 45 s.t s.t PROPN bracis-19045 96 46 . . PROPN bracis-19045 96 47 } } PUNCT bracis-19045 96 48 \quad \quad PROPN bracis-19045 96 49 \vec \vec PROPN bracis-19045 96 50 { { PUNCT bracis-19045 96 51 1}^t 1}^t PROPN bracis-19045 96 52 \vec \vec PROPN bracis-19045 96 53 { { PUNCT bracis-19045 96 54 w}_i w}_i PROPN bracis-19045 96 55 = = SYM bracis-19045 96 56 1 1 NUM bracis-19045 96 57 \end{aligned}$$ \end{aligned}$$ NOUN bracis-19045 96 58 ( ( PUNCT bracis-19045 96 59 15 15 NUM bracis-19045 96 60 ) ) PUNCT bracis-19045 96 61 for for ADP bracis-19045 96 62 \(i \(i NOUN bracis-19045 96 63 = = SYM bracis-19045 96 64 1 1 NUM bracis-19045 96 65 , , PUNCT bracis-19045 96 66 2 2 NUM bracis-19045 96 67 , , PUNCT bracis-19045 96 68 ... ... PUNCT bracis-19045 96 69 , , PUNCT bracis-19045 96 70 n\ n\ NOUN bracis-19045 96 71 ) ) PUNCT bracis-19045 96 72 . . PUNCT bracis-19045 97 1 the the DET bracis-19045 97 2 lagrangian lagrangian ADJ bracis-19045 97 3 function function NOUN bracis-19045 97 4 is be AUX bracis-19045 97 5 defined define VERB bracis-19045 97 6 by by ADP bracis-19045 97 7 : : PUNCT bracis-19045 97 8 $ $ SYM bracis-19045 97 9 $ $ SYM bracis-19045 97 10 \begin{aligned \begin{aligne VERB bracis-19045 97 11 } } PUNCT bracis-19045 97 12 l(\vec l(\vec X bracis-19045 97 13 { { PUNCT bracis-19045 97 14 w}_i w}_i PROPN bracis-19045 97 15 , , PUNCT bracis-19045 97 16 \lambda \lambda PROPN bracis-19045 97 17 ) ) PUNCT bracis-19045 98 1 = = SYM bracis-19045 98 2 \vec \vec PROPN bracis-19045 98 3 { { PUNCT bracis-19045 98 4 w}_i^t w}_i^t X bracis-19045 98 5 c_i c_i SYM bracis-19045 98 6 \vec \vec NOUN bracis-19045 98 7 { { PUNCT bracis-19045 98 8 w}_i w}_i PROPN bracis-19045 98 9 + + SYM bracis-19045 98 10 \alpha \alpha PROPN bracis-19045 98 11 \vec \vec PROPN bracis-19045 98 12 { { PUNCT bracis-19045 98 13 w}_i^t\vec w}_i^t\vec PROPN bracis-19045 98 14 { { PUNCT bracis-19045 98 15 w}_i w}_i NUM bracis-19045 98 16 \lambda \lambda PROPN bracis-19045 98 17 ( ( PUNCT bracis-19045 98 18 \vec \vec PROPN bracis-19045 98 19 { { PUNCT bracis-19045 98 20 1}^t 1}^t PROPN bracis-19045 98 21 \vec \vec PROPN bracis-19045 98 22 { { PUNCT bracis-19045 98 23 w}_i w}_i PROPN bracis-19045 98 24 1 1 NUM bracis-19045 98 25 ) ) PUNCT bracis-19045 98 26 \end{aligned}$$ \end{aligned}$$ X bracis-19045 98 27 ( ( PUNCT bracis-19045 98 28 16 16 NUM bracis-19045 98 29 ) ) PUNCT bracis-19045 98 30 taking take VERB bracis-19045 98 31 the the DET bracis-19045 98 32 derivative derivative NOUN bracis-19045 98 33 with with ADP bracis-19045 98 34 respect respect NOUN bracis-19045 98 35 to to ADP bracis-19045 98 36 \(\vec \(\vec NOUN bracis-19045 98 37 { { PUNCT bracis-19045 98 38 w}_i\ w}_i\ NOUN bracis-19045 98 39 ) ) PUNCT bracis-19045 98 40 and and CCONJ bracis-19045 98 41 setting set VERB bracis-19045 98 42 the the DET bracis-19045 98 43 result result NOUN bracis-19045 98 44 to to ADP bracis-19045 98 45 zero zero NUM bracis-19045 98 46 : : PUNCT bracis-19045 98 47 $ $ SYM bracis-19045 98 48 $ $ SYM bracis-19045 98 49 \begin{aligned \begin{aligne VERB bracis-19045 98 50 } } PUNCT bracis-19045 98 51 2 2 NUM bracis-19045 98 52 c_i c_i SYM bracis-19045 98 53 \vec \vec NOUN bracis-19045 98 54 { { PUNCT bracis-19045 98 55 w}_i w}_i X bracis-19045 98 56 + + CCONJ bracis-19045 98 57 2 2 NUM bracis-19045 98 58 \alpha \alpha PROPN bracis-19045 98 59 \vec \vec PROPN bracis-19045 98 60 { { PUNCT bracis-19045 98 61 w}_i w}_i PROPN bracis-19045 98 62 = = SYM bracis-19045 98 63 \lambda \lambda PROPN bracis-19045 98 64 \vec \vec PROPN bracis-19045 98 65 { { PUNCT bracis-19045 98 66 1 1 NUM bracis-19045 98 67 } } PUNCT bracis-19045 98 68 \end{aligned}$$ \end{aligned}$$ X bracis-19045 98 69 ( ( PUNCT bracis-19045 98 70 17 17 NUM bracis-19045 98 71 ) ) PUNCT bracis-19045 98 72 $ $ SYM bracis-19045 98 73 $ $ SYM bracis-19045 98 74 \begin{aligned \begin{aligne VERB bracis-19045 98 75 } } PUNCT bracis-19045 98 76 ( ( PUNCT bracis-19045 98 77 c_i c_i PUNCT bracis-19045 99 1 + + CCONJ bracis-19045 99 2 \alpha \alpha ADJ bracis-19045 99 3 i)\vec i)\vec PROPN bracis-19045 99 4 { { PUNCT bracis-19045 99 5 w}_i w}_i PROPN bracis-19045 99 6 = = SYM bracis-19045 99 7 \frac{\lambda \frac{\lambda NOUN bracis-19045 99 8 } } PUNCT bracis-19045 99 9 { { PUNCT bracis-19045 99 10 2}\vec 2}\vec NUM bracis-19045 99 11 { { PUNCT bracis-19045 99 12 1 1 NUM bracis-19045 99 13 } } PUNCT bracis-19045 99 14 \end{aligned}$$ \end{aligned}$$ X bracis-19045 99 15 ( ( PUNCT bracis-19045 99 16 18 18 NUM bracis-19045 99 17 ) ) PUNCT bracis-19045 99 18 $ $ SYM bracis-19045 99 19 $ $ SYM bracis-19045 99 20 \begin{aligned \begin{aligne VERB bracis-19045 99 21 } } PUNCT bracis-19045 99 22 \vec \vec PROPN bracis-19045 99 23 { { PUNCT bracis-19045 99 24 w}_i w}_i PROPN bracis-19045 99 25 = = SYM bracis-19045 99 26 \frac{\lambda \frac{\lambda NOUN bracis-19045 99 27 } } PUNCT bracis-19045 99 28 { { PUNCT bracis-19045 99 29 2}(c_i 2}(c_i NUM bracis-19045 99 30 + + CCONJ bracis-19045 99 31 \alpha \alpha ADJ bracis-19045 99 32 i)^{-1 i)^{-1 NOUN bracis-19045 99 33 } } PUNCT bracis-19045 99 34 \vec \vec PROPN bracis-19045 99 35 { { PUNCT bracis-19045 99 36 1 1 NUM bracis-19045 99 37 } } PUNCT bracis-19045 99 38 \end{aligned}$$ \end{aligned}$$ X bracis-19045 99 39 ( ( PUNCT bracis-19045 99 40 19 19 NUM bracis-19045 99 41 ) ) PUNCT bracis-19045 99 42 where where SCONJ bracis-19045 99 43 \(\lambda \(\lambda NOUN bracis-19045 99 44 \ \ NOUN bracis-19045 99 45 ) ) PUNCT bracis-19045 99 46 is be AUX bracis-19045 99 47 selected select VERB bracis-19045 99 48 to to PART bracis-19045 99 49 properly properly ADV bracis-19045 99 50 normalize normalize VERB bracis-19045 99 51 \(\vec \(\vec NOUN bracis-19045 99 52 { { PUNCT bracis-19045 99 53 w}_i\ w}_i\ NOUN bracis-19045 99 54 ) ) PUNCT bracis-19045 99 55 , , PUNCT bracis-19045 99 56 which which PRON bracis-19045 99 57 is be AUX bracis-19045 99 58 equivalent equivalent ADJ bracis-19045 99 59 to to ADP bracis-19045 99 60 solving solve VERB bracis-19045 99 61 the the DET bracis-19045 99 62 following follow VERB bracis-19045 99 63 linear linear ADJ bracis-19045 99 64 system system NOUN bracis-19045 99 65 : : PUNCT bracis-19045 99 66 $ $ SYM bracis-19045 99 67 $ $ SYM bracis-19045 99 68 \begin{aligned \begin{aligne VERB bracis-19045 99 69 } } PUNCT bracis-19045 99 70 ( ( PUNCT bracis-19045 99 71 c_i c_i PUNCT bracis-19045 99 72 + + PUNCT bracis-19045 99 73 \alpha \alpha ADJ bracis-19045 99 74 i i NOUN bracis-19045 99 75 ) ) PUNCT bracis-19045 99 76 \vec \vec PROPN bracis-19045 99 77 { { PUNCT bracis-19045 99 78 w}_i w}_i PROPN bracis-19045 99 79 = = SYM bracis-19045 99 80 \vec \vec PROPN bracis-19045 99 81 { { PUNCT bracis-19045 99 82 1 1 NUM bracis-19045 99 83 } } PUNCT bracis-19045 99 84 \end{aligned}$$ \end{aligned}$$ X bracis-19045 99 85 ( ( PUNCT bracis-19045 99 86 20 20 NUM bracis-19045 99 87 ) ) PUNCT bracis-19045 99 88 and and CCONJ bracis-19045 99 89 then then ADV bracis-19045 99 90 normalize normalize VERB bracis-19045 99 91 the the DET bracis-19045 99 92 solution solution NOUN bracis-19045 99 93 . . PUNCT bracis-19045 100 1 in in ADP bracis-19045 100 2 other other ADJ bracis-19045 100 3 words word NOUN bracis-19045 100 4 , , PUNCT bracis-19045 100 5 to to PART bracis-19045 100 6 regularize regularize VERB bracis-19045 100 7 the the DET bracis-19045 100 8 problem problem NOUN bracis-19045 100 9 it it PRON bracis-19045 100 10 is be AUX bracis-19045 100 11 necessary necessary ADJ bracis-19045 100 12 to to PART bracis-19045 100 13 add add VERB bracis-19045 100 14 a a DET bracis-19045 100 15 small small ADJ bracis-19045 100 16 perturbation perturbation NOUN bracis-19045 100 17 into into ADP bracis-19045 100 18 the the DET bracis-19045 100 19 main main ADJ bracis-19045 100 20 diagonal diagonal NOUN bracis-19045 100 21 of of ADP bracis-19045 100 22 the the DET bracis-19045 100 23 matrix matrix NOUN bracis-19045 100 24 \(c_i\ \(c_i\ NOUN bracis-19045 100 25 ) ) PUNCT bracis-19045 100 26 . . PUNCT bracis-19045 101 1 in in ADP bracis-19045 101 2 all all DET bracis-19045 101 3 experiments experiment NOUN bracis-19045 101 4 reported report VERB bracis-19045 101 5 in in ADP bracis-19045 101 6 the the DET bracis-19045 101 7 present present ADJ bracis-19045 101 8 paper paper NOUN bracis-19045 101 9 , , PUNCT bracis-19045 101 10 we we PRON bracis-19045 101 11 use use VERB bracis-19045 101 12 the the DET bracis-19045 101 13 regularization regularization NOUN bracis-19045 101 14 parameter parameter NOUN bracis-19045 101 15 \(\alpha \(\alpha NOUN bracis-19045 101 16 = = SYM bracis-19045 101 17 10^{-4}\ 10^{-4}\ NUM bracis-19045 101 18 ) ) PUNCT bracis-19045 101 19 . . PUNCT bracis-19045 102 1 finding find VERB bracis-19045 102 2 the the DET bracis-19045 102 3 coordinates coordinate NOUN bracis-19045 102 4 . . PUNCT bracis-19045 103 1 the the DET bracis-19045 103 2 main main ADJ bracis-19045 103 3 idea idea NOUN bracis-19045 103 4 in in ADP bracis-19045 103 5 the the DET bracis-19045 103 6 last last ADJ bracis-19045 103 7 stage stage NOUN bracis-19045 103 8 of of ADP bracis-19045 103 9 the the DET bracis-19045 103 10 lle lle PROPN bracis-19045 103 11 algorithm algorithm NOUN bracis-19045 103 12 is be AUX bracis-19045 103 13 to to PART bracis-19045 103 14 use use VERB bracis-19045 103 15 the the DET bracis-19045 103 16 optimal optimal ADJ bracis-19045 103 17 reconstruction reconstruction NOUN bracis-19045 103 18 weights weight NOUN bracis-19045 103 19 estimated estimate VERB bracis-19045 103 20 by by ADP bracis-19045 103 21 least least ADJ bracis-19045 103 22 - - PUNCT bracis-19045 103 23 squares square NOUN bracis-19045 103 24 as as ADP bracis-19045 103 25 the the DET bracis-19045 103 26 proper proper ADJ bracis-19045 103 27 weights weight NOUN bracis-19045 103 28 on on ADP bracis-19045 103 29 the the DET bracis-19045 103 30 manifold manifold NOUN bracis-19045 103 31 and and CCONJ bracis-19045 103 32 then then ADV bracis-19045 103 33 solve solve VERB bracis-19045 103 34 for for ADP bracis-19045 103 35 the the DET bracis-19045 103 36 local local ADJ bracis-19045 103 37 manifold manifold ADJ bracis-19045 103 38 coordinates coordinate NOUN bracis-19045 103 39 . . PUNCT bracis-19045 104 1 thus thus ADV bracis-19045 104 2 , , PUNCT bracis-19045 104 3 fixing fix VERB bracis-19045 104 4 the the DET bracis-19045 104 5 weight weight NOUN bracis-19045 104 6 matrix matrix NOUN bracis-19045 104 7 w w PROPN bracis-19045 104 8 , , PUNCT bracis-19045 104 9 the the DET bracis-19045 104 10 goal goal NOUN bracis-19045 104 11 is be AUX bracis-19045 104 12 to to PART bracis-19045 104 13 solve solve VERB bracis-19045 104 14 another another DET bracis-19045 104 15 quadratic quadratic ADJ bracis-19045 104 16 minimization minimization NOUN bracis-19045 104 17 problem problem NOUN bracis-19045 104 18 to to PART bracis-19045 104 19 minimize minimize VERB bracis-19045 104 20 the the DET bracis-19045 104 21 following following NOUN bracis-19045 104 22 : : PUNCT bracis-19045 104 23 $ $ SYM bracis-19045 104 24 $ $ SYM bracis-19045 104 25 \begin{aligned \begin{aligne VERB bracis-19045 104 26 } } PUNCT bracis-19045 104 27 \varphi \varphi PROPN bracis-19045 104 28 ( ( PUNCT bracis-19045 104 29 y y NOUN bracis-19045 104 30 ) ) PUNCT bracis-19045 104 31 = = VERB bracis-19045 104 32 \sum \sum NOUN bracis-19045 104 33 _ _ PUNCT bracis-19045 104 34 { { PUNCT bracis-19045 104 35 i=1}^{n i=1}^{n ADJ bracis-19045 104 36 } } PUNCT bracis-19045 104 37 \left\vert \left\vert PROPN bracis-19045 104 38 \vec \vec PROPN bracis-19045 104 39 { { PUNCT bracis-19045 104 40 y}_i y}_i PROPN bracis-19045 104 41 \sum \sum NOUN bracis-19045 104 42 _ _ PUNCT bracis-19045 104 43 j j PROPN bracis-19045 104 44 w_{ij w_{ij PROPN bracis-19045 104 45 } } PUNCT bracis-19045 104 46 \vec \vec PROPN bracis-19045 104 47 { { PUNCT bracis-19045 104 48 y}_j y}_j PROPN bracis-19045 104 49 \right\vert \right\vert PROPN bracis-19045 104 50 ^2 ^2 X bracis-19045 104 51 \end{aligned}$$ \end{aligned}$$ X bracis-19045 104 52 ( ( PUNCT bracis-19045 104 53 21 21 NUM bracis-19045 104 54 ) ) PUNCT bracis-19045 104 55 hence hence ADV bracis-19045 104 56 , , PUNCT bracis-19045 104 57 the the DET bracis-19045 104 58 following follow VERB bracis-19045 104 59 question question NOUN bracis-19045 104 60 needs need VERB bracis-19045 104 61 to to PART bracis-19045 104 62 be be AUX bracis-19045 104 63 addressed address VERB bracis-19045 104 64 : : PUNCT bracis-19045 104 65 what what PRON bracis-19045 104 66 are be AUX bracis-19045 104 67 the the DET bracis-19045 104 68 coordinates coordinate NOUN bracis-19045 104 69 \(\vec \(\vec PUNCT bracis-19045 104 70 { { PUNCT bracis-19045 104 71 y}_i y}_i PROPN bracis-19045 104 72 \in \in PROPN bracis-19045 104 73 r^d\ r^d\ PROPN bracis-19045 104 74 ) ) PUNCT bracis-19045 104 75 ( ( PUNCT bracis-19045 104 76 approximately approximately ADV bracis-19045 104 77 on on ADP bracis-19045 104 78 the the DET bracis-19045 104 79 manifold manifold NOUN bracis-19045 104 80 ) ) PUNCT bracis-19045 104 81 , , PUNCT bracis-19045 104 82 that that SCONJ bracis-19045 104 83 such such ADJ bracis-19045 104 84 weights weight NOUN bracis-19045 104 85 ( ( PUNCT bracis-19045 104 86 w w NOUN bracis-19045 104 87 ) ) PUNCT bracis-19045 104 88 reconstruct reconstruct NOUN bracis-19045 104 89 ? ? PUNCT bracis-19045 105 1 in in ADP bracis-19045 105 2 order order NOUN bracis-19045 105 3 to to PART bracis-19045 105 4 avoid avoid VERB bracis-19045 105 5 degeneracy degeneracy NOUN bracis-19045 105 6 , , PUNCT bracis-19045 105 7 we we PRON bracis-19045 105 8 have have VERB bracis-19045 105 9 to to PART bracis-19045 105 10 impose impose VERB bracis-19045 105 11 two two NUM bracis-19045 105 12 constraints constraint NOUN bracis-19045 105 13 , , PUNCT bracis-19045 105 14 as as SCONJ bracis-19045 105 15 follows follow VERB bracis-19045 105 16 : : PUNCT bracis-19045 105 17 1 1 X bracis-19045 105 18 . . PUNCT bracis-19045 106 1 the the DET bracis-19045 106 2 mean mean NOUN bracis-19045 106 3 of of ADP bracis-19045 106 4 the the DET bracis-19045 106 5 data datum NOUN bracis-19045 106 6 in in ADP bracis-19045 106 7 the the DET bracis-19045 106 8 transformed transform VERB bracis-19045 106 9 space space NOUN bracis-19045 106 10 is be AUX bracis-19045 106 11 zero zero NUM bracis-19045 106 12 , , PUNCT bracis-19045 106 13 otherwise otherwise ADV bracis-19045 106 14 we we PRON bracis-19045 106 15 would would AUX bracis-19045 106 16 have have VERB bracis-19045 106 17 an an DET bracis-19045 106 18 infinite infinite ADJ bracis-19045 106 19 number number NOUN bracis-19045 106 20 of of ADP bracis-19045 106 21 solutions solution NOUN bracis-19045 106 22 ; ; PUNCT bracis-19045 106 23 2 2 X bracis-19045 106 24 . . X bracis-19045 106 25 the the DET bracis-19045 106 26 covariance covariance NOUN bracis-19045 106 27 matrix matrix NOUN bracis-19045 106 28 of of ADP bracis-19045 106 29 the the DET bracis-19045 106 30 transformed transform VERB bracis-19045 106 31 data data NOUN bracis-19045 106 32 is be AUX bracis-19045 106 33 the the DET bracis-19045 106 34 identity identity NOUN bracis-19045 106 35 matrix matrix NOUN bracis-19045 106 36 , , PUNCT bracis-19045 106 37 therefore therefore ADV bracis-19045 106 38 , , PUNCT bracis-19045 106 39 there there PRON bracis-19045 106 40 is be VERB bracis-19045 106 41 no no DET bracis-19045 106 42 correlation correlation NOUN bracis-19045 106 43 between between ADP bracis-19045 106 44 the the DET bracis-19045 106 45 components component NOUN bracis-19045 106 46 of of ADP bracis-19045 106 47 \(\vec \(\vec X bracis-19045 106 48 { { PUNCT bracis-19045 106 49 y y PROPN bracis-19045 106 50 } } PUNCT bracis-19045 106 51 \in \in PROPN bracis-19045 106 52 r^d\ r^d\ NOUN bracis-19045 106 53 ) ) PUNCT bracis-19045 106 54 ( ( PUNCT bracis-19045 106 55 this this PRON bracis-19045 106 56 is be AUX bracis-19045 106 57 a a DET bracis-19045 106 58 statistical statistical ADJ bracis-19045 106 59 constraint constraint NOUN bracis-19045 106 60 to to PART bracis-19045 106 61 assess assess VERB bracis-19045 106 62 that that SCONJ bracis-19045 106 63 the the DET bracis-19045 106 64 output output NOUN bracis-19045 106 65 space space NOUN bracis-19045 106 66 is be AUX bracis-19045 106 67 the the DET bracis-19045 106 68 euclidean euclidean ADJ bracis-19045 106 69 one one NOUN bracis-19045 106 70 , , PUNCT bracis-19045 106 71 defined define VERB bracis-19045 106 72 by by ADP bracis-19045 106 73 an an DET bracis-19045 106 74 orthogonal orthogonal ADJ bracis-19045 106 75 basis basis NOUN bracis-19045 106 76 ) ) PUNCT bracis-19045 106 77 . . PUNCT bracis-19045 107 1 however however ADV bracis-19045 107 2 , , PUNCT bracis-19045 107 3 unlikely unlikely ADJ bracis-19045 107 4 the the DET bracis-19045 107 5 estimation estimation NOUN bracis-19045 107 6 of of ADP bracis-19045 107 7 the the DET bracis-19045 107 8 weights weight NOUN bracis-19045 107 9 w w NOUN bracis-19045 107 10 , , PUNCT bracis-19045 107 11 finding find VERB bracis-19045 107 12 the the DET bracis-19045 107 13 coordinates coordinate NOUN bracis-19045 107 14 does do AUX bracis-19045 107 15 not not PART bracis-19045 107 16 simplify simplify VERB bracis-19045 107 17 into into ADP bracis-19045 107 18 n n CCONJ bracis-19045 107 19 independent independent ADJ bracis-19045 107 20 problems problem NOUN bracis-19045 107 21 , , PUNCT bracis-19045 107 22 because because SCONJ bracis-19045 107 23 each each DET bracis-19045 107 24 row row NOUN bracis-19045 107 25 of of ADP bracis-19045 107 26 y y PROPN bracis-19045 107 27 appears appear VERB bracis-19045 107 28 in in ADP bracis-19045 107 29 \(\varphi \(\varphi ADJ bracis-19045 107 30 \ \ NOUN bracis-19045 107 31 ) ) PUNCT bracis-19045 107 32 multiple multiple ADJ bracis-19045 107 33 times time NOUN bracis-19045 107 34 , , PUNCT bracis-19045 107 35 as as ADP bracis-19045 107 36 the the DET bracis-19045 107 37 central central ADJ bracis-19045 107 38 vector vector NOUN bracis-19045 107 39 \(y_i\ \(y_i\ NOUN bracis-19045 107 40 ) ) PUNCT bracis-19045 107 41 and and CCONJ bracis-19045 107 42 also also ADV bracis-19045 107 43 as as ADP bracis-19045 107 44 one one NUM bracis-19045 107 45 of of ADP bracis-19045 107 46 the the DET bracis-19045 107 47 neighbors neighbor NOUN bracis-19045 107 48 of of ADP bracis-19045 107 49 other other ADJ bracis-19045 107 50 vectors vector NOUN bracis-19045 107 51 . . PUNCT bracis-19045 108 1 thus thus ADV bracis-19045 108 2 , , PUNCT bracis-19045 108 3 firstly firstly ADV bracis-19045 108 4 , , PUNCT bracis-19045 108 5 we we PRON bracis-19045 108 6 rewrite rewrite VERB bracis-19045 108 7 equation equation NOUN bracis-19045 108 8 ( ( PUNCT bracis-19045 108 9 21 21 NUM bracis-19045 108 10 ) ) PUNCT bracis-19045 108 11 in in ADP bracis-19045 108 12 a a DET bracis-19045 108 13 more more ADV bracis-19045 108 14 meaningful meaningful ADJ bracis-19045 108 15 manner manner NOUN bracis-19045 108 16 using use VERB bracis-19045 108 17 matrices matrix NOUN bracis-19045 108 18 , , PUNCT bracis-19045 108 19 as as SCONJ bracis-19045 108 20 follows follow VERB bracis-19045 108 21 : : PUNCT bracis-19045 108 22 $ $ SYM bracis-19045 108 23 $ $ SYM bracis-19045 108 24 \begin{aligned \begin{aligne VERB bracis-19045 108 25 } } PUNCT bracis-19045 108 26 \varphi \varphi PROPN bracis-19045 108 27 ( ( PUNCT bracis-19045 108 28 y y NOUN bracis-19045 108 29 ) ) PUNCT bracis-19045 108 30 = = VERB bracis-19045 108 31 \sum \sum NOUN bracis-19045 108 32 _ _ PUNCT bracis-19045 108 33 { { PUNCT bracis-19045 108 34 i=1}^{n i=1}^{n ADJ bracis-19045 108 35 } } PUNCT bracis-19045 108 36 \left \left PROPN bracis-19045 108 37 ( ( PUNCT bracis-19045 108 38 \vec \vec PROPN bracis-19045 108 39 { { PUNCT bracis-19045 108 40 y}_i y}_i PROPN bracis-19045 108 41 \sum \sum NOUN bracis-19045 109 1 _ _ PUNCT bracis-19045 109 2 j j PROPN bracis-19045 109 3 w_{ij w_{ij PROPN bracis-19045 109 4 } } PUNCT bracis-19045 109 5 \vec \vec PROPN bracis-19045 109 6 { { PUNCT bracis-19045 109 7 y}_j y}_j PROPN bracis-19045 109 8 \right \right PROPN bracis-19045 109 9 ) ) PUNCT bracis-19045 109 10 ^t ^t PROPN bracis-19045 109 11 \left \left PROPN bracis-19045 109 12 ( ( PUNCT bracis-19045 109 13 \vec \vec PROPN bracis-19045 109 14 { { PUNCT bracis-19045 109 15 y}_i y}_i PROPN bracis-19045 109 16 \sum \sum NOUN bracis-19045 110 1 _ _ PUNCT bracis-19045 110 2 j j PROPN bracis-19045 110 3 w_{ij w_{ij PROPN bracis-19045 110 4 } } PUNCT bracis-19045 110 5 \vec \vec PROPN bracis-19045 110 6 { { PUNCT bracis-19045 110 7 y}_j y}_j PROPN bracis-19045 110 8 \right \right PROPN bracis-19045 110 9 ) ) PUNCT bracis-19045 110 10 \end{aligned}$$ \end{aligned}$$ X bracis-19045 110 11 ( ( PUNCT bracis-19045 110 12 22 22 NUM bracis-19045 110 13 ) ) PUNCT bracis-19045 110 14 applying apply VERB bracis-19045 110 15 the the DET bracis-19045 110 16 distributive distributive ADJ bracis-19045 110 17 law law NOUN bracis-19045 110 18 and and CCONJ bracis-19045 110 19 expanding expand VERB bracis-19045 110 20 the the DET bracis-19045 110 21 summation summation NOUN bracis-19045 110 22 , , PUNCT bracis-19045 110 23 we we PRON bracis-19045 110 24 then then ADV bracis-19045 110 25 obtain obtain VERB bracis-19045 110 26 : : PUNCT bracis-19045 110 27 $ $ SYM bracis-19045 110 28 $ $ SYM bracis-19045 110 29 \begin{aligned \begin{aligne VERB bracis-19045 110 30 } } PUNCT bracis-19045 110 31 \varphi \varphi PROPN bracis-19045 110 32 ( ( PUNCT bracis-19045 110 33 y)&= y)&= NUM bracis-19045 110 34 \sum \sum PROPN bracis-19045 110 35 _ _ PUNCT bracis-19045 110 36 { { PUNCT bracis-19045 110 37 i=1}^{n i=1}^{n ADJ bracis-19045 110 38 } } PUNCT bracis-19045 110 39 \vec \vec PROPN bracis-19045 110 40 { { PUNCT bracis-19045 110 41 y}_i^t\vec y}_i^t\vec PROPN bracis-19045 110 42 { { PUNCT bracis-19045 110 43 y}_i y}_i PROPN bracis-19045 110 44 \sum \sum NOUN bracis-19045 111 1 _ _ PUNCT bracis-19045 111 2 { { PUNCT bracis-19045 111 3 i=1}^{n i=1}^{n ADJ bracis-19045 111 4 } } PUNCT bracis-19045 111 5 \sum \sum NOUN bracis-19045 112 1 _ _ PUNCT bracis-19045 112 2 j j PROPN bracis-19045 112 3 \vec \vec PROPN bracis-19045 112 4 { { PUNCT bracis-19045 112 5 y}_i^t y}_i^t NOUN bracis-19045 112 6 w_{ij w_{ij PROPN bracis-19045 112 7 } } PUNCT bracis-19045 112 8 \vec \vec PROPN bracis-19045 112 9 { { PUNCT bracis-19045 112 10 y}_j y}_j NOUN bracis-19045 112 11 \nonumber \nonumber AUX bracis-19045 112 12 \\&\sum \\&\sum VERB bracis-19045 112 13 _ _ PRON bracis-19045 112 14 { { PUNCT bracis-19045 112 15 i=1}^{n}\sum i=1}^{n}\sum ADP bracis-19045 112 16 _ _ PUNCT bracis-19045 112 17 j j X bracis-19045 112 18 \vec \vec PROPN bracis-19045 112 19 { { PUNCT bracis-19045 112 20 y}_j^t y}_j^t PROPN bracis-19045 112 21 w_{ji w_{ji PROPN bracis-19045 112 22 } } PUNCT bracis-19045 112 23 \vec \vec PROPN bracis-19045 112 24 { { PUNCT bracis-19045 112 25 y}_i y}_i PROPN bracis-19045 112 26 + + NUM bracis-19045 112 27 \sum \sum NOUN bracis-19045 112 28 _ _ PUNCT bracis-19045 112 29 { { PUNCT bracis-19045 112 30 i=1}^{n i=1}^{n ADJ bracis-19045 112 31 } } PUNCT bracis-19045 112 32 \sum \sum NOUN bracis-19045 113 1 _ _ PUNCT bracis-19045 114 1 j j PROPN bracis-19045 114 2 \sum \sum NOUN bracis-19045 114 3 _ _ PUNCT bracis-19045 114 4 k k PROPN bracis-19045 114 5 \vec \vec PROPN bracis-19045 114 6 { { PUNCT bracis-19045 114 7 y}_j^t y}_j^t PROPN bracis-19045 114 8 w_{ji w_{ji PROPN bracis-19045 114 9 } } PUNCT bracis-19045 114 10 w_{ik w_{ik PROPN bracis-19045 114 11 } } PUNCT bracis-19045 114 12 \vec \vec PROPN bracis-19045 114 13 { { PUNCT bracis-19045 114 14 y}_k y}_k PROPN bracis-19045 114 15 \end{aligned}$$ \end{aligned}$$ X bracis-19045 114 16 ( ( PUNCT bracis-19045 114 17 23 23 NUM bracis-19045 114 18 ) ) PUNCT bracis-19045 114 19 denoting denote VERB bracis-19045 114 20 by by ADP bracis-19045 114 21 y y PRON bracis-19045 114 22 the the DET bracis-19045 114 23 \(d \(d NOUN bracis-19045 114 24 \times \times ADP bracis-19045 114 25 n\ n\ NOUN bracis-19045 114 26 ) ) PUNCT bracis-19045 114 27 matrix matrix NOUN bracis-19045 114 28 in in ADP bracis-19045 114 29 which which PRON bracis-19045 114 30 each each DET bracis-19045 114 31 column column NOUN bracis-19045 114 32 \(\vec \(\vec X bracis-19045 114 33 { { PUNCT bracis-19045 114 34 y}_i\ y}_i\ NOUN bracis-19045 114 35 ) ) PUNCT bracis-19045 114 36 for for ADP bracis-19045 114 37 \(i=1,2, \(i=1,2, NOUN bracis-19045 114 38 ... ... PUNCT bracis-19045 114 39 ,n\ ,n\ PUNCT bracis-19045 114 40 ) ) PUNCT bracis-19045 115 1 stores store VERB bracis-19045 115 2 the the DET bracis-19045 115 3 coordinates coordinate NOUN bracis-19045 115 4 of of ADP bracis-19045 115 5 the the DET bracis-19045 115 6 i i PROPN bracis-19045 115 7 - - PUNCT bracis-19045 115 8 th th X bracis-19045 115 9 sample sample NOUN bracis-19045 115 10 in in ADP bracis-19045 115 11 the the DET bracis-19045 115 12 manifold manifold NOUN bracis-19045 115 13 and and CCONJ bracis-19045 115 14 acknowledging acknowledge VERB bracis-19045 115 15 that that SCONJ bracis-19045 115 16 \(\vec \(\vec NOUN bracis-19045 115 17 { { PUNCT bracis-19045 115 18 w}_i(j w}_i(j X bracis-19045 115 19 ) ) PUNCT bracis-19045 115 20 = = SYM bracis-19045 115 21 0\ 0\ PROPN bracis-19045 115 22 ) ) PUNCT bracis-19045 115 23 , , PUNCT bracis-19045 115 24 unless unless SCONJ bracis-19045 115 25 \(\vec \(\vec NOUN bracis-19045 115 26 { { PUNCT bracis-19045 115 27 y}_j\ y}_j\ PROPN bracis-19045 115 28 ) ) PUNCT bracis-19045 115 29 is be AUX bracis-19045 115 30 one one NUM bracis-19045 115 31 of of ADP bracis-19045 115 32 the the DET bracis-19045 115 33 neighbors neighbor NOUN bracis-19045 115 34 of of ADP bracis-19045 115 35 \(\vec \(\vec X bracis-19045 115 36 { { PUNCT bracis-19045 115 37 y}_i\ y}_i\ NOUN bracis-19045 115 38 ) ) PUNCT bracis-19045 115 39 , , PUNCT bracis-19045 115 40 we we PRON bracis-19045 115 41 may may AUX bracis-19045 115 42 express express VERB bracis-19045 115 43 \(\varphi \(\varphi PROPN bracis-19045 115 44 ( ( PUNCT bracis-19045 115 45 y)\ y)\ NOUN bracis-19045 115 46 ) ) PUNCT bracis-19045 115 47 as as SCONJ bracis-19045 115 48 follows follow VERB bracis-19045 115 49 : : PUNCT bracis-19045 116 1 $ $ SYM bracis-19045 116 2 $ $ SYM bracis-19045 116 3 \begin{aligned \begin{aligne VERB bracis-19045 116 4 } } PUNCT bracis-19045 116 5 \varphi \varphi PROPN bracis-19045 116 6 ( ( PUNCT bracis-19045 116 7 y)&= y)&= X bracis-19045 116 8 tr(y^t tr(y^t VERB bracis-19045 116 9 y y PROPN bracis-19045 116 10 ) ) PUNCT bracis-19045 116 11 tr(y^t tr(y^t NOUN bracis-19045 116 12 w w PROPN bracis-19045 116 13 y y NOUN bracis-19045 116 14 ) ) PUNCT bracis-19045 116 15 \nonumber \nonumber ADP bracis-19045 116 16 \\&\qquad \\&\qquad PRON bracis-19045 116 17 tr(y^t tr(y^t NOUN bracis-19045 116 18 w^t w^t VERB bracis-19045 116 19 y y PROPN bracis-19045 116 20 ) ) PUNCT bracis-19045 116 21 + + NUM bracis-19045 116 22 tr(y^t tr(y^t NOUN bracis-19045 116 23 w^t w^t VERB bracis-19045 116 24 w w PROPN bracis-19045 116 25 y y PROPN bracis-19045 116 26 ) ) PUNCT bracis-19045 116 27 \nonumber \nonumber PROPN bracis-19045 116 28 \\&= \\&= NOUN bracis-19045 116 29 tr(y^t tr(y^t NOUN bracis-19045 116 30 y y NOUN bracis-19045 116 31 ) ) PUNCT bracis-19045 116 32 tr(y^t(wy tr(y^t(wy NOUN bracis-19045 116 33 ) ) PUNCT bracis-19045 116 34 ) ) PUNCT bracis-19045 117 1 \nonumber \nonumber ADP bracis-19045 117 2 \\&\qquad \\&\qquad PROPN bracis-19045 117 3 tr((wy)^t tr((wy)^t PROPN bracis-19045 117 4 y y PROPN bracis-19045 117 5 ) ) PUNCT bracis-19045 117 6 + + NUM bracis-19045 117 7 tr((wy)^t(wy tr((wy)^t(wy PROPN bracis-19045 117 8 ) ) PUNCT bracis-19045 117 9 ) ) PUNCT bracis-19045 118 1 \nonumber \nonumber PROPN bracis-19045 118 2 \\&= \\&= PROPN bracis-19045 118 3 tr(y^t(y tr(y^t(y PROPN bracis-19045 118 4 wy wy PROPN bracis-19045 118 5 ) ) PUNCT bracis-19045 118 6 ( ( PUNCT bracis-19045 118 7 wy)^t(y wy)^t(y X bracis-19045 118 8 wy wy PROPN bracis-19045 118 9 ) ) PUNCT bracis-19045 118 10 ) ) PUNCT bracis-19045 118 11 \nonumber \nonumber PROPN bracis-19045 118 12 \\&= \\&= PROPN bracis-19045 118 13 tr((y tr((y NOUN bracis-19045 118 14 wy)^t wy)^t NOUN bracis-19045 118 15 ( ( PUNCT bracis-19045 118 16 y y PROPN bracis-19045 118 17 wy wy PROPN bracis-19045 118 18 ) ) PUNCT bracis-19045 118 19 ) ) PUNCT bracis-19045 119 1 \nonumber \nonumber PROPN bracis-19045 119 2 \\&= \\&= X bracis-19045 119 3 tr(((i tr(((i NOUN bracis-19045 119 4 - - PUNCT bracis-19045 119 5 w)y)^t w)y)^t NOUN bracis-19045 119 6 ( ( PUNCT bracis-19045 119 7 ( ( PUNCT bracis-19045 119 8 i i PRON bracis-19045 119 9 - - PUNCT bracis-19045 119 10 w)y w)y ADV bracis-19045 119 11 ) ) PUNCT bracis-19045 119 12 ) ) PUNCT bracis-19045 120 1 \nonumber \nonumber PROPN bracis-19045 120 2 \\&= \\&= NOUN bracis-19045 120 3 tr(y^t tr(y^t NOUN bracis-19045 120 4 ( ( PUNCT bracis-19045 120 5 i i PRON bracis-19045 120 6 w)^t w)^t PROPN bracis-19045 120 7 ( ( PUNCT bracis-19045 120 8 i i NOUN bracis-19045 120 9 w w PROPN bracis-19045 120 10 ) ) PUNCT bracis-19045 120 11 y y PROPN bracis-19045 120 12 ) ) PUNCT bracis-19045 120 13 \end{aligned}$$ \end{aligned}$$ X bracis-19045 120 14 ( ( PUNCT bracis-19045 120 15 24 24 NUM bracis-19045 120 16 ) ) PUNCT bracis-19045 120 17 defining define VERB bracis-19045 120 18 the the DET bracis-19045 120 19 \(n \(n NOUN bracis-19045 120 20 \times \times ADP bracis-19045 120 21 n\ n\ NOUN bracis-19045 120 22 ) ) PUNCT bracis-19045 120 23 matrix matrix NOUN bracis-19045 120 24 m m VERB bracis-19045 120 25 as as ADP bracis-19045 120 26 : : PUNCT bracis-19045 120 27 $ $ SYM bracis-19045 120 28 $ $ SYM bracis-19045 120 29 \begin{aligned \begin{aligne VERB bracis-19045 120 30 } } PUNCT bracis-19045 120 31 m m VERB bracis-19045 120 32 = = PUNCT bracis-19045 120 33 ( ( PUNCT bracis-19045 121 1 i i PRON bracis-19045 121 2 w)^t w)^t PROPN bracis-19045 121 3 ( ( PUNCT bracis-19045 121 4 i i NOUN bracis-19045 121 5 w w PROPN bracis-19045 121 6 ) ) PUNCT bracis-19045 121 7 \end{aligned}$$ \end{aligned}$$ X bracis-19045 121 8 ( ( PUNCT bracis-19045 121 9 25 25 NUM bracis-19045 121 10 ) ) PUNCT bracis-19045 121 11 we we PRON bracis-19045 121 12 get get VERB bracis-19045 121 13 the the DET bracis-19045 121 14 following follow VERB bracis-19045 121 15 optimization optimization NOUN bracis-19045 121 16 problem problem NOUN bracis-19045 121 17 : : PUNCT bracis-19045 122 1 $ $ SYM bracis-19045 122 2 $ $ SYM bracis-19045 122 3 \begin{aligned \begin{aligne VERB bracis-19045 122 4 } } PUNCT bracis-19045 122 5 \mathop \mathop PROPN bracis-19045 122 6 { { PUNCT bracis-19045 122 7 \mathrm \mathrm PROPN bracis-19045 122 8 { { PUNCT bracis-19045 122 9 arg\,min}}\limits arg\,min}}\limits X bracis-19045 122 10 _ _ X bracis-19045 122 11 { { PUNCT bracis-19045 122 12 y}~ y}~ PROPN bracis-19045 122 13 tr(y^t tr(y^t VERB bracis-19045 122 14 m m VERB bracis-19045 122 15 y y NOUN bracis-19045 122 16 ) ) PUNCT bracis-19045 122 17 \quad \quad PROPN bracis-19045 122 18 \text \text NOUN bracis-19045 122 19 { { PUNCT bracis-19045 122 20 subject subject ADJ bracis-19045 122 21 to to ADP bracis-19045 122 22 } } PUNCT bracis-19045 122 23 \quad \quad PROPN bracis-19045 122 24 \frac{1}{n}y^t \frac{1}{n}y^t X bracis-19045 122 25 y y PROPN bracis-19045 122 26 = = PROPN bracis-19045 122 27 i i PRON bracis-19045 122 28 \end{aligned}$$ \end{aligned}$$ X bracis-19045 122 29 ( ( PUNCT bracis-19045 122 30 26 26 NUM bracis-19045 122 31 ) ) PUNCT bracis-19045 122 32 thus thus ADV bracis-19045 122 33 , , PUNCT bracis-19045 122 34 the the DET bracis-19045 122 35 lagrangian lagrangian ADJ bracis-19045 122 36 function function NOUN bracis-19045 122 37 is be AUX bracis-19045 122 38 given give VERB bracis-19045 122 39 by by ADP bracis-19045 122 40 : : PUNCT bracis-19045 122 41 $ $ SYM bracis-19045 122 42 $ $ SYM bracis-19045 122 43 \begin{aligned \begin{aligne VERB bracis-19045 122 44 } } PUNCT bracis-19045 122 45 l(y l(y PROPN bracis-19045 122 46 , , PUNCT bracis-19045 122 47 \lambda \lambda PROPN bracis-19045 122 48 ) ) PUNCT bracis-19045 123 1 = = SYM bracis-19045 123 2 tr(y^t tr(y^t NOUN bracis-19045 123 3 m m VERB bracis-19045 123 4 y y NOUN bracis-19045 123 5 ) ) PUNCT bracis-19045 124 1 \lambda \lambda PROPN bracis-19045 124 2 \left \left PROPN bracis-19045 124 3 ( ( PUNCT bracis-19045 124 4 \frac{1}{n}y^t \frac{1}{n}y^t NOUN bracis-19045 124 5 y y PROPN bracis-19045 124 6 i i PRON bracis-19045 124 7 \right \right PROPN bracis-19045 124 8 ) ) PUNCT bracis-19045 124 9 \end{aligned}$$ \end{aligned}$$ X bracis-19045 124 10 ( ( PUNCT bracis-19045 124 11 27 27 NUM bracis-19045 124 12 ) ) PUNCT bracis-19045 124 13 differentiating differentiate VERB bracis-19045 124 14 and and CCONJ bracis-19045 124 15 setting set VERB bracis-19045 124 16 the the DET bracis-19045 124 17 result result NOUN bracis-19045 124 18 to to ADP bracis-19045 124 19 zero zero NUM bracis-19045 124 20 , , PUNCT bracis-19045 124 21 finally finally ADV bracis-19045 124 22 leads lead VERB bracis-19045 124 23 to to ADP bracis-19045 124 24 : : PUNCT bracis-19045 124 25 $ $ SYM bracis-19045 124 26 $ $ SYM bracis-19045 124 27 \begin{aligned \begin{aligne VERB bracis-19045 124 28 } } PUNCT bracis-19045 124 29 my my PRON bracis-19045 124 30 = = PUNCT bracis-19045 124 31 \beta \beta PROPN bracis-19045 124 32 y y PROPN bracis-19045 124 33 \end{aligned}$$ \end{aligned}$$ X bracis-19045 124 34 ( ( PUNCT bracis-19045 124 35 28 28 NUM bracis-19045 124 36 ) ) PUNCT bracis-19045 124 37 where where SCONJ bracis-19045 124 38 \(\beta \(\beta NOUN bracis-19045 124 39 = = NOUN bracis-19045 124 40 \frac{\lambda \frac{\lambda PRON bracis-19045 124 41 } } PUNCT bracis-19045 124 42 { { PUNCT bracis-19045 124 43 n}\ n}\ NUM bracis-19045 124 44 ) ) PUNCT bracis-19045 124 45 , , PUNCT bracis-19045 124 46 showing show VERB bracis-19045 124 47 that that SCONJ bracis-19045 124 48 the the DET bracis-19045 124 49 y y PROPN bracis-19045 124 50 must must AUX bracis-19045 124 51 be be AUX bracis-19045 124 52 composed compose VERB bracis-19045 124 53 by by ADP bracis-19045 124 54 the the DET bracis-19045 124 55 eigenvectors eigenvector NOUN bracis-19045 124 56 of of ADP bracis-19045 124 57 the the DET bracis-19045 124 58 matrix matrix NOUN bracis-19045 124 59 m. m. NOUN bracis-19045 124 60 since since SCONJ bracis-19045 124 61 there there PRON bracis-19045 124 62 is be VERB bracis-19045 124 63 a a DET bracis-19045 124 64 minimization minimization NOUN bracis-19045 124 65 problem problem NOUN bracis-19045 124 66 to to PART bracis-19045 124 67 be be AUX bracis-19045 124 68 solved solve VERB bracis-19045 124 69 , , PUNCT bracis-19045 124 70 it it PRON bracis-19045 124 71 is be AUX bracis-19045 124 72 then then ADV bracis-19045 124 73 required require VERB bracis-19045 124 74 to to PART bracis-19045 124 75 select select VERB bracis-19045 124 76 y y PROPN bracis-19045 124 77 to to PART bracis-19045 124 78 compose compose VERB bracis-19045 124 79 the the DET bracis-19045 124 80 d d PROPN bracis-19045 124 81 eigenvectors eigenvector NOUN bracis-19045 124 82 associated associate VERB bracis-19045 124 83 to to ADP bracis-19045 124 84 the the DET bracis-19045 124 85 d d PROPN bracis-19045 124 86 smallest small ADJ bracis-19045 124 87 eigenvalues eigenvalue NOUN bracis-19045 124 88 . . PUNCT bracis-19045 125 1 note note VERB bracis-19045 125 2 that that SCONJ bracis-19045 125 3 m m VERB bracis-19045 125 4 being be AUX bracis-19045 125 5 an an DET bracis-19045 125 6 \(n \(n NOUN bracis-19045 125 7 \times \times ADP bracis-19045 125 8 n\ n\ NOUN bracis-19045 125 9 ) ) PUNCT bracis-19045 125 10 matrix matrix NOUN bracis-19045 125 11 , , PUNCT bracis-19045 125 12 it it PRON bracis-19045 125 13 contains contain VERB bracis-19045 125 14 n n PRON bracis-19045 125 15 eigenvalues eigenvalue NOUN bracis-19045 125 16 and and CCONJ bracis-19045 125 17 n n PRON bracis-19045 125 18 orthogonal orthogonal ADJ bracis-19045 125 19 eigenvectors eigenvector NOUN bracis-19045 125 20 . . PUNCT bracis-19045 126 1 although although SCONJ bracis-19045 126 2 the the DET bracis-19045 126 3 eigenvalues eigenvalue NOUN bracis-19045 126 4 are be AUX bracis-19045 126 5 real real ADJ bracis-19045 126 6 and and CCONJ bracis-19045 126 7 non non ADJ bracis-19045 126 8 - - ADJ bracis-19045 126 9 negative negative ADJ bracis-19045 126 10 values value NOUN bracis-19045 126 11 , , PUNCT bracis-19045 126 12 its its PRON bracis-19045 126 13 smallest small ADJ bracis-19045 126 14 value value NOUN bracis-19045 126 15 is be AUX bracis-19045 126 16 always always ADV bracis-19045 126 17 zero zero NUM bracis-19045 126 18 , , PUNCT bracis-19045 126 19 with with ADP bracis-19045 126 20 the the DET bracis-19045 126 21 constant constant ADJ bracis-19045 126 22 eigenvector eigenvector NOUN bracis-19045 126 23 \(\vec \(\vec NOUN bracis-19045 126 24 { { PUNCT bracis-19045 126 25 1}\ 1}\ NUM bracis-19045 126 26 ) ) PUNCT bracis-19045 126 27 . . PUNCT bracis-19045 127 1 such such DET bracis-19045 127 2 a a DET bracis-19045 127 3 bottom bottom ADJ bracis-19045 127 4 eigenvector eigenvector NOUN bracis-19045 127 5 corresponds correspond NOUN bracis-19045 127 6 to to ADP bracis-19045 127 7 the the DET bracis-19045 127 8 mean mean NOUN bracis-19045 127 9 of of ADP bracis-19045 127 10 y y PROPN bracis-19045 127 11 and and CCONJ bracis-19045 127 12 should should AUX bracis-19045 127 13 be be AUX bracis-19045 127 14 discarded discard VERB bracis-19045 127 15 to to PART bracis-19045 127 16 impose impose VERB bracis-19045 127 17 the the DET bracis-19045 127 18 constraint constraint NOUN bracis-19045 127 19 that that PRON bracis-19045 127 20 \(\sum \(\sum VERB bracis-19045 128 1 _ _ PRON bracis-19045 128 2 { { PUNCT bracis-19045 128 3 i=1}^{n i=1}^{n ADJ bracis-19045 128 4 } } PUNCT bracis-19045 128 5 \vec \vec PROPN bracis-19045 128 6 { { PUNCT bracis-19045 128 7 y}_i y}_i PROPN bracis-19045 128 8 = = SYM bracis-19045 128 9 0\ 0\ PROPN bracis-19045 128 10 ) ) PUNCT bracis-19045 129 1 [ [ X bracis-19045 129 2 7 7 NUM bracis-19045 129 3 ] ] PUNCT bracis-19045 129 4 . . PUNCT bracis-19045 130 1 therefore therefore ADV bracis-19045 130 2 , , PUNCT bracis-19045 130 3 to to PART bracis-19045 130 4 get get VERB bracis-19045 130 5 \(\vec \(\vec NOUN bracis-19045 130 6 { { PUNCT bracis-19045 130 7 y}_i y}_i PROPN bracis-19045 130 8 \in \in PROPN bracis-19045 130 9 r^d\ r^d\ PROPN bracis-19045 130 10 ) ) PUNCT bracis-19045 130 11 , , PUNCT bracis-19045 130 12 where where SCONJ bracis-19045 130 13 \(d \(d NOUN bracis-19045 130 14 < < X bracis-19045 130 15 m\ m\ NOUN bracis-19045 130 16 ) ) PUNCT bracis-19045 130 17 , , PUNCT bracis-19045 130 18 it it PRON bracis-19045 130 19 is be AUX bracis-19045 130 20 necessary necessary ADJ bracis-19045 130 21 to to PART bracis-19045 130 22 select select VERB bracis-19045 130 23 the the DET bracis-19045 130 24 \(d+1\ \(d+1\ NOUN bracis-19045 130 25 ) ) PUNCT bracis-19045 130 26 smallest small ADJ bracis-19045 130 27 eigenvectors eigenvector NOUN bracis-19045 130 28 and and CCONJ bracis-19045 130 29 discard discard VERB bracis-19045 130 30 the the DET bracis-19045 130 31 constant constant ADJ bracis-19045 130 32 eigenvector eigenvector NOUN bracis-19045 130 33 with with ADP bracis-19045 130 34 zero zero NUM bracis-19045 130 35 eigenvalue eigenvalue NOUN bracis-19045 130 36 . . PUNCT bracis-19045 131 1 in in ADP bracis-19045 131 2 other other ADJ bracis-19045 131 3 words word NOUN bracis-19045 131 4 , , PUNCT bracis-19045 131 5 one one PRON bracis-19045 131 6 must must AUX bracis-19045 131 7 select select VERB bracis-19045 131 8 the the DET bracis-19045 131 9 d d PROPN bracis-19045 131 10 eigenvectors eigenvector NOUN bracis-19045 131 11 associated associate VERB bracis-19045 131 12 to to ADP bracis-19045 131 13 the the DET bracis-19045 131 14 bottom bottom ADJ bracis-19045 131 15 non non ADJ bracis-19045 131 16 - - ADJ bracis-19045 131 17 zero zero NUM bracis-19045 131 18 eigenvalues eigenvalue NOUN bracis-19045 131 19 . . PUNCT bracis-19045 132 1 3 3 NUM bracis-19045 132 2 the the DET bracis-19045 132 3 kl kl PROPN bracis-19045 132 4 divergence divergence NOUN bracis-19045 132 5 - - PUNCT bracis-19045 132 6 based base VERB bracis-19045 132 7 lle lle NOUN bracis-19045 132 8 method method VERB bracis-19045 132 9 the the DET bracis-19045 132 10 main main ADJ bracis-19045 132 11 motivation motivation NOUN bracis-19045 132 12 to to PART bracis-19045 132 13 introduce introduce VERB bracis-19045 132 14 the the DET bracis-19045 132 15 lle lle PROPN bracis-19045 132 16 - - PUNCT bracis-19045 132 17 kl kl PROPN bracis-19045 132 18 method method NOUN bracis-19045 132 19 is be AUX bracis-19045 132 20 to to PART bracis-19045 132 21 find find VERB bracis-19045 132 22 a a DET bracis-19045 132 23 surrogate surrogate NOUN bracis-19045 132 24 for for ADP bracis-19045 132 25 the the DET bracis-19045 132 26 local local ADJ bracis-19045 132 27 matrix matrix NOUN bracis-19045 132 28 \(c_i\ \(c_i\ NOUN bracis-19045 132 29 ) ) PUNCT bracis-19045 132 30 for for ADP bracis-19045 132 31 each each DET bracis-19045 132 32 sample sample NOUN bracis-19045 132 33 of of ADP bracis-19045 132 34 the the DET bracis-19045 132 35 dataset dataset NOUN bracis-19045 132 36 . . PUNCT bracis-19045 133 1 recall recall VERB bracis-19045 133 2 that that PRON bracis-19045 133 3 , , PUNCT bracis-19045 133 4 originally originally ADV bracis-19045 133 5 , , PUNCT bracis-19045 133 6 \(c_i(j \(c_i(j NUM bracis-19045 133 7 , , PUNCT bracis-19045 133 8 k)\ k)\ NOUN bracis-19045 133 9 ) ) PUNCT bracis-19045 133 10 is be AUX bracis-19045 133 11 computed compute VERB bracis-19045 133 12 as as ADP bracis-19045 133 13 the the DET bracis-19045 133 14 inner inner ADJ bracis-19045 133 15 product product NOUN bracis-19045 133 16 between between ADP bracis-19045 133 17 \(\vec \(\vec NOUN bracis-19045 133 18 { { PUNCT bracis-19045 133 19 x}_i x}_i PROPN bracis-19045 133 20 \vec \vec PROPN bracis-19045 133 21 { { PUNCT bracis-19045 133 22 x}_j\ x}_j\ PROPN bracis-19045 133 23 ) ) PUNCT bracis-19045 133 24 and and CCONJ bracis-19045 133 25 \(\vec \(\vec NOUN bracis-19045 133 26 { { PUNCT bracis-19045 133 27 x}_i x}_i PROPN bracis-19045 133 28 \vec \vec PROPN bracis-19045 133 29 { { PUNCT bracis-19045 133 30 x}_k\ x}_k\ PROPN bracis-19045 133 31 ) ) PUNCT bracis-19045 133 32 , , PUNCT bracis-19045 133 33 which which PRON bracis-19045 133 34 means mean VERB bracis-19045 133 35 that that SCONJ bracis-19045 133 36 we we PRON bracis-19045 133 37 employ employ VERB bracis-19045 133 38 the the DET bracis-19045 133 39 euclidean euclidean ADJ bracis-19045 133 40 geometry geometry NOUN bracis-19045 133 41 in in ADP bracis-19045 133 42 the the DET bracis-19045 133 43 estimation estimation NOUN bracis-19045 133 44 of of ADP bracis-19045 133 45 the the DET bracis-19045 133 46 optimal optimal ADJ bracis-19045 133 47 reconstruction reconstruction NOUN bracis-19045 133 48 weights weight NOUN bracis-19045 133 49 . . PUNCT bracis-19045 134 1 in in ADP bracis-19045 134 2 the the DET bracis-19045 134 3 definition definition NOUN bracis-19045 134 4 of of ADP bracis-19045 134 5 such such ADJ bracis-19045 134 6 matrix matrix NOUN bracis-19045 134 7 it it PRON bracis-19045 134 8 is be AUX bracis-19045 134 9 used use VERB bracis-19045 134 10 a a DET bracis-19045 134 11 non non ADJ bracis-19045 134 12 - - ADJ bracis-19045 134 13 linear linear ADJ bracis-19045 134 14 distance distance NOUN bracis-19045 134 15 function function NOUN bracis-19045 134 16 , , PUNCT bracis-19045 134 17 namely namely ADV bracis-19045 134 18 the the DET bracis-19045 134 19 relative relative ADJ bracis-19045 134 20 entropy entropy NOUN bracis-19045 134 21 between between ADP bracis-19045 134 22 gaussian gaussian ADJ bracis-19045 134 23 densities density NOUN bracis-19045 134 24 estimated estimate VERB bracis-19045 134 25 within within ADP bracis-19045 134 26 different different ADJ bracis-19045 134 27 patches patch NOUN bracis-19045 134 28 of of ADP bracis-19045 134 29 the the DET bracis-19045 134 30 knn knn NOUN bracis-19045 134 31 graph graph NOUN bracis-19045 134 32 . . PUNCT bracis-19045 135 1 our our PRON bracis-19045 135 2 inspiration inspiration NOUN bracis-19045 135 3 is be AUX bracis-19045 135 4 the the DET bracis-19045 135 5 parametric parametric ADJ bracis-19045 135 6 pca pca PROPN bracis-19045 135 7 , , PUNCT bracis-19045 135 8 an an DET bracis-19045 135 9 information information NOUN bracis-19045 135 10 - - PUNCT bracis-19045 135 11 theoretic theoretic NOUN bracis-19045 135 12 extension extension NOUN bracis-19045 135 13 of of ADP bracis-19045 135 14 the the DET bracis-19045 135 15 pca pca NOUN bracis-19045 135 16 method method NOUN bracis-19045 135 17 that that PRON bracis-19045 135 18 applies apply VERB bracis-19045 135 19 the the DET bracis-19045 135 20 kl kl PROPN bracis-19045 135 21 divergence divergence NOUN bracis-19045 135 22 to to PART bracis-19045 135 23 compute compute VERB bracis-19045 135 24 a a DET bracis-19045 135 25 surrogate surrogate NOUN bracis-19045 135 26 for for ADP bracis-19045 135 27 the the DET bracis-19045 135 28 covariance covariance NOUN bracis-19045 135 29 matrix matrix NOUN bracis-19045 135 30 ( ( PUNCT bracis-19045 135 31 i.e. i.e. X bracis-19045 135 32 , , PUNCT bracis-19045 135 33 entropic entropic ADJ bracis-19045 135 34 covariance covariance NOUN bracis-19045 135 35 matrix matrix NOUN bracis-19045 135 36 ) ) PUNCT bracis-19045 136 1 [ [ X bracis-19045 136 2 5 5 NUM bracis-19045 136 3 ] ] PUNCT bracis-19045 136 4 . . PUNCT bracis-19045 137 1 let let VERB bracis-19045 137 2 \(x \(x NOUN bracis-19045 137 3 = = SYM bracis-19045 137 4 \ \ ADJ bracis-19045 137 5 { { PUNCT bracis-19045 137 6 \vec \vec PROPN bracis-19045 137 7 { { PUNCT bracis-19045 137 8 x}_1 x}_1 PROPN bracis-19045 137 9 , , PUNCT bracis-19045 137 10 \vec \vec PROPN bracis-19045 137 11 { { PUNCT bracis-19045 137 12 x}_2 x}_2 PROPN bracis-19045 137 13 , , PUNCT bracis-19045 137 14 \ldots \ldots INTJ bracis-19045 137 15 , , PUNCT bracis-19045 137 16 \vec \vec PROPN bracis-19045 137 17 { { PUNCT bracis-19045 137 18 x}_n x}_n PROPN bracis-19045 137 19 \}\ \}\ NOUN bracis-19045 137 20 ) ) PUNCT bracis-19045 137 21 , , PUNCT bracis-19045 137 22 with with ADP bracis-19045 137 23 \(\vec \(\vec NOUN bracis-19045 137 24 { { PUNCT bracis-19045 137 25 x}_i x}_i PROPN bracis-19045 137 26 \in \in PROPN bracis-19045 137 27 r^m\ r^m\ NOUN bracis-19045 137 28 ) ) PUNCT bracis-19045 137 29 , , PUNCT bracis-19045 137 30 be be AUX bracis-19045 137 31 our our PRON bracis-19045 137 32 data datum NOUN bracis-19045 137 33 matrix matrix NOUN bracis-19045 137 34 . . PUNCT bracis-19045 138 1 the the DET bracis-19045 138 2 first first ADJ bracis-19045 138 3 step step NOUN bracis-19045 138 4 in in ADP bracis-19045 138 5 the the DET bracis-19045 138 6 proposed propose VERB bracis-19045 138 7 method method NOUN bracis-19045 138 8 consists consist VERB bracis-19045 138 9 of of ADP bracis-19045 138 10 building build VERB bracis-19045 138 11 the the DET bracis-19045 138 12 knn knn NOUN bracis-19045 138 13 graph graph NOUN bracis-19045 138 14 from from ADP bracis-19045 138 15 x. x. NOUN bracis-19045 138 16 at at ADP bracis-19045 138 17 this this DET bracis-19045 138 18 early early ADJ bracis-19045 138 19 stage stage NOUN bracis-19045 138 20 , , PUNCT bracis-19045 138 21 we we PRON bracis-19045 138 22 employ employ VERB bracis-19045 138 23 the the DET bracis-19045 138 24 extrinsic extrinsic ADJ bracis-19045 138 25 euclidean euclidean ADJ bracis-19045 138 26 distance distance NOUN bracis-19045 138 27 to to PART bracis-19045 138 28 compute compute VERB bracis-19045 138 29 the the DET bracis-19045 138 30 nearest near ADJ bracis-19045 138 31 neighbors neighbor NOUN bracis-19045 138 32 of of ADP bracis-19045 138 33 each each DET bracis-19045 138 34 sample sample NOUN bracis-19045 138 35 \(\vec \(\vec NOUN bracis-19045 138 36 { { PUNCT bracis-19045 138 37 x}_i\ x}_i\ NOUN bracis-19045 138 38 ) ) PUNCT bracis-19045 138 39 . . PUNCT bracis-19045 139 1 denoting denote VERB bracis-19045 139 2 by by ADP bracis-19045 139 3 \(\eta \(\eta NOUN bracis-19045 139 4 _ _ NOUN bracis-19045 139 5 i\ i\ NOUN bracis-19045 139 6 ) ) PUNCT bracis-19045 139 7 the the DET bracis-19045 139 8 neighborhood neighborhood NOUN bracis-19045 139 9 system system NOUN bracis-19045 139 10 of of ADP bracis-19045 139 11 \(\vec \(\vec X bracis-19045 139 12 { { PUNCT bracis-19045 139 13 x}_i\ x}_i\ NOUN bracis-19045 139 14 ) ) PUNCT bracis-19045 139 15 , , PUNCT bracis-19045 139 16 a a DET bracis-19045 139 17 patch patch NOUN bracis-19045 139 18 \(p_i\ \(p_i\ NOUN bracis-19045 139 19 ) ) PUNCT bracis-19045 139 20 is be AUX bracis-19045 139 21 defined define VERB bracis-19045 139 22 as as ADP bracis-19045 139 23 the the DET bracis-19045 139 24 set set NOUN bracis-19045 139 25 \(\ \(\ ADP bracis-19045 139 26 { { PUNCT bracis-19045 139 27 \vec \vec PROPN bracis-19045 139 28 { { PUNCT bracis-19045 139 29 x}_i x}_i PROPN bracis-19045 139 30 \cup \cup VERB bracis-19045 139 31 \eta \eta PROPN bracis-19045 139 32 _ _ NOUN bracis-19045 139 33 i i PRON bracis-19045 139 34 \}\ \}\ NOUN bracis-19045 139 35 ) ) PUNCT bracis-19045 139 36 . . PUNCT bracis-19045 140 1 it it PRON bracis-19045 140 2 is be AUX bracis-19045 140 3 worth worth ADJ bracis-19045 140 4 noticing notice VERB bracis-19045 140 5 that that SCONJ bracis-19045 140 6 the the DET bracis-19045 140 7 number number NOUN bracis-19045 140 8 of of ADP bracis-19045 140 9 elements element NOUN bracis-19045 140 10 of of ADP bracis-19045 140 11 \(p_i\ \(p_i\ NOUN bracis-19045 140 12 ) ) PUNCT bracis-19045 140 13 is be AUX bracis-19045 140 14 \(k+1\ \(k+1\ NOUN bracis-19045 140 15 ) ) PUNCT bracis-19045 140 16 , , PUNCT bracis-19045 140 17 for for ADP bracis-19045 140 18 \(i \(i NOUN bracis-19045 140 19 = = SYM bracis-19045 140 20 1 1 NUM bracis-19045 140 21 , , PUNCT bracis-19045 140 22 2 2 NUM bracis-19045 140 23 , , PUNCT bracis-19045 140 24 ... ... PUNCT bracis-19045 140 25 , , PUNCT bracis-19045 140 26 n\ n\ NOUN bracis-19045 140 27 ) ) PUNCT bracis-19045 140 28 . . PUNCT bracis-19045 141 1 in in ADP bracis-19045 141 2 other other ADJ bracis-19045 141 3 words word NOUN bracis-19045 141 4 , , PUNCT bracis-19045 141 5 a a DET bracis-19045 141 6 patch patch NOUN bracis-19045 141 7 \(p_i\ \(p_i\ NOUN bracis-19045 141 8 ) ) PUNCT bracis-19045 141 9 is be AUX bracis-19045 141 10 given give VERB bracis-19045 141 11 by by ADP bracis-19045 141 12 an an DET bracis-19045 141 13 \(m \(m NOUN bracis-19045 141 14 \times \times PROPN bracis-19045 141 15 ( ( PUNCT bracis-19045 141 16 k+1)\ k+1)\ PROPN bracis-19045 141 17 ) ) PUNCT bracis-19045 141 18 matrix matrix NOUN bracis-19045 141 19 : : PUNCT bracis-19045 141 20 $ $ SYM bracis-19045 141 21 $ $ SYM bracis-19045 141 22 \begin{aligned \begin{aligne VERB bracis-19045 141 23 } } PUNCT bracis-19045 141 24 p_i p_i SYM bracis-19045 141 25 = = SYM bracis-19045 141 26 \begin{bmatrix \begin{bmatrix X bracis-19045 141 27 } } PUNCT bracis-19045 141 28 x_i(1 x_i(1 NUM bracis-19045 141 29 ) ) PUNCT bracis-19045 141 30 & & CCONJ bracis-19045 141 31 { { PUNCT bracis-19045 141 32 } } PUNCT bracis-19045 141 33 x_{i1}(1 x_{i1}(1 PROPN bracis-19045 141 34 ) ) PUNCT bracis-19045 141 35 & & CCONJ bracis-19045 141 36 { { PUNCT bracis-19045 141 37 } } PUNCT bracis-19045 141 38 \ldots \ldot NOUN bracis-19045 141 39 & & CCONJ bracis-19045 141 40 { { PUNCT bracis-19045 141 41 } } PUNCT bracis-19045 141 42 x_{ik}(1 x_{ik}(1 PROPN bracis-19045 141 43 ) ) PUNCT bracis-19045 141 44 \\ \\ NOUN bracis-19045 141 45 x_i(2 x_i(2 PUNCT bracis-19045 141 46 ) ) PUNCT bracis-19045 141 47 & & CCONJ bracis-19045 141 48 { { PUNCT bracis-19045 141 49 } } PUNCT bracis-19045 141 50 x_{i1}(2 x_{i1}(2 PROPN bracis-19045 141 51 ) ) PUNCT bracis-19045 141 52 & & CCONJ bracis-19045 141 53 { { PUNCT bracis-19045 141 54 } } PUNCT bracis-19045 141 55 \ldots \ldot NOUN bracis-19045 141 56 & & CCONJ bracis-19045 141 57 { { PUNCT bracis-19045 141 58 } } PUNCT bracis-19045 141 59 x_{ik}(2 x_{ik}(2 PROPN bracis-19045 141 60 ) ) PUNCT bracis-19045 141 61 \\ \\ NOUN bracis-19045 141 62 \vdots \vdots PROPN bracis-19045 141 63 & & CCONJ bracis-19045 141 64 { { PUNCT bracis-19045 141 65 } } PUNCT bracis-19045 141 66 \vdots \vdots PROPN bracis-19045 141 67 & & CCONJ bracis-19045 141 68 { { PUNCT bracis-19045 141 69 } } PUNCT bracis-19045 141 70 \ddots \ddot NOUN bracis-19045 141 71 & & CCONJ bracis-19045 141 72 { { PUNCT bracis-19045 141 73 } } PUNCT bracis-19045 141 74 \vdots \vdot NOUN bracis-19045 141 75 \\ \\ NOUN bracis-19045 141 76 \vdots \vdot NOUN bracis-19045 141 77 & & CCONJ bracis-19045 141 78 { { PUNCT bracis-19045 141 79 } } PUNCT bracis-19045 141 80 \vdots \vdot NOUN bracis-19045 141 81 & & CCONJ bracis-19045 141 82 { { PUNCT bracis-19045 141 83 } } PUNCT bracis-19045 141 84 \ldots \ldots PROPN bracis-19045 141 85 & & CCONJ bracis-19045 141 86 { { PUNCT bracis-19045 141 87 } } PUNCT bracis-19045 141 88 \vdots \vdot NOUN bracis-19045 141 89 \\ \\ NOUN bracis-19045 141 90 x_i(m x_i(m NUM bracis-19045 141 91 ) ) PUNCT bracis-19045 141 92 & & CCONJ bracis-19045 141 93 { { PUNCT bracis-19045 141 94 } } PUNCT bracis-19045 141 95 x_{i1}(m x_{i1}(m PROPN bracis-19045 141 96 ) ) PUNCT bracis-19045 141 97 & & CCONJ bracis-19045 141 98 { { PUNCT bracis-19045 141 99 } } PUNCT bracis-19045 141 100 \ldots \ldots PROPN bracis-19045 141 101 & & CCONJ bracis-19045 141 102 { { PUNCT bracis-19045 141 103 } } PUNCT bracis-19045 141 104 x_{ik}(m x_{ik}(m ADJ bracis-19045 141 105 ) ) PUNCT bracis-19045 141 106 \end{bmatrix \end{bmatrix NOUN bracis-19045 141 107 } } PUNCT bracis-19045 141 108 \end{aligned}$$ \end{aligned}$$ X bracis-19045 141 109 ( ( PUNCT bracis-19045 141 110 29 29 NUM bracis-19045 141 111 ) ) PUNCT bracis-19045 141 112 the the DET bracis-19045 141 113 rationale rationale NOUN bracis-19045 141 114 of of ADP bracis-19045 141 115 the the DET bracis-19045 141 116 proposed propose VERB bracis-19045 141 117 method method NOUN bracis-19045 141 118 is be AUX bracis-19045 141 119 considering consider VERB bracis-19045 141 120 each each DET bracis-19045 141 121 row row NOUN bracis-19045 141 122 of of ADP bracis-19045 141 123 the the DET bracis-19045 141 124 matrix matrix NOUN bracis-19045 141 125 \(p_i\ \(p_i\ NOUN bracis-19045 141 126 ) ) PUNCT bracis-19045 141 127 as as ADP bracis-19045 141 128 a a DET bracis-19045 141 129 sample sample NOUN bracis-19045 141 130 of of ADP bracis-19045 141 131 a a DET bracis-19045 141 132 univariate univariate ADJ bracis-19045 141 133 gaussian gaussian ADJ bracis-19045 141 134 random random ADJ bracis-19045 141 135 variable variable NOUN bracis-19045 141 136 , , PUNCT bracis-19045 141 137 and and CCONJ bracis-19045 141 138 then then ADV bracis-19045 141 139 estimating estimate VERB bracis-19045 141 140 the the DET bracis-19045 141 141 parameters parameter NOUN bracis-19045 141 142 \(\mu \(\mu ADP bracis-19045 141 143 \ \ NOUN bracis-19045 141 144 ) ) PUNCT bracis-19045 141 145 ( ( PUNCT bracis-19045 141 146 mean mean VERB bracis-19045 141 147 ) ) PUNCT bracis-19045 141 148 and and CCONJ bracis-19045 141 149 \(\sigma \(\sigma NOUN bracis-19045 141 150 ^2\ ^2\ NOUN bracis-19045 141 151 ) ) PUNCT bracis-19045 141 152 ( ( PUNCT bracis-19045 141 153 variance variance NOUN bracis-19045 141 154 ) ) PUNCT bracis-19045 141 155 of of ADP bracis-19045 141 156 each each DET bracis-19045 141 157 row row NOUN bracis-19045 141 158 , , PUNCT bracis-19045 141 159 leading lead VERB bracis-19045 141 160 to to ADP bracis-19045 141 161 an an DET bracis-19045 141 162 m m ADV bracis-19045 141 163 - - ADJ bracis-19045 141 164 dimensional dimensional ADJ bracis-19045 141 165 vector vector NOUN bracis-19045 141 166 of of ADP bracis-19045 141 167 tuples tuple NOUN bracis-19045 141 168 , , PUNCT bracis-19045 141 169 as as SCONJ bracis-19045 141 170 follows follow VERB bracis-19045 141 171 : : PUNCT bracis-19045 141 172 $ $ SYM bracis-19045 141 173 $ $ SYM bracis-19045 141 174 \begin{aligned \begin{aligne VERB bracis-19045 141 175 } } PUNCT bracis-19045 141 176 \vec \vec PROPN bracis-19045 141 177 { { PUNCT bracis-19045 141 178 p}_i p}_i PROPN bracis-19045 141 179 = = SYM bracis-19045 141 180 \left \left PROPN bracis-19045 141 181 [ [ PUNCT bracis-19045 141 182 ( ( PUNCT bracis-19045 141 183 \mu \mu PROPN bracis-19045 141 184 _ _ SYM bracis-19045 141 185 i(1 i(1 NOUN bracis-19045 141 186 ) ) PUNCT bracis-19045 141 187 , , PUNCT bracis-19045 141 188 \sigma \sigma PROPN bracis-19045 141 189 _ _ PROPN bracis-19045 141 190 i^2(1 i^2(1 NOUN bracis-19045 141 191 ) ) PUNCT bracis-19045 141 192 ) ) PUNCT bracis-19045 141 193 , , PUNCT bracis-19045 141 194 ... ... PUNCT bracis-19045 141 195 , , PUNCT bracis-19045 141 196 ( ( PUNCT bracis-19045 141 197 \mu \mu NOUN bracis-19045 141 198 _ _ NOUN bracis-19045 141 199 i(m i(m NOUN bracis-19045 141 200 ) ) PUNCT bracis-19045 141 201 , , PUNCT bracis-19045 141 202 \sigma \sigma PROPN bracis-19045 141 203 _ _ NOUN bracis-19045 141 204 i^2(m i^2(m NOUN bracis-19045 141 205 ) ) PUNCT bracis-19045 141 206 ) ) PUNCT bracis-19045 142 1 \right \right ADP bracis-19045 142 2 ] ] X bracis-19045 142 3 \end{aligned}$$ \end{aligned}$$ X bracis-19045 142 4 ( ( PUNCT bracis-19045 142 5 30 30 NUM bracis-19045 142 6 ) ) PUNCT bracis-19045 142 7 let let VERB bracis-19045 142 8 p(x p(x NOUN bracis-19045 142 9 ) ) PUNCT bracis-19045 142 10 and and CCONJ bracis-19045 142 11 q(x q(x PROPN bracis-19045 142 12 ) ) PUNCT bracis-19045 142 13 be be VERB bracis-19045 142 14 univariate univariate ADJ bracis-19045 142 15 gaussian gaussian ADJ bracis-19045 142 16 densities density NOUN bracis-19045 142 17 , , PUNCT bracis-19045 142 18 \(n(\mu \(n(\mu NOUN bracis-19045 142 19 _ _ NOUN bracis-19045 142 20 1 1 NUM bracis-19045 142 21 , , PUNCT bracis-19045 142 22 \sigma \sigma PROPN bracis-19045 142 23 _ _ NOUN bracis-19045 142 24 1 1 NUM bracis-19045 142 25 ^ ^ SYM bracis-19045 142 26 2)\ 2)\ NUM bracis-19045 142 27 ) ) PUNCT bracis-19045 142 28 and and CCONJ bracis-19045 142 29 \(n(\mu \(n(\mu VERB bracis-19045 142 30 _ _ NOUN bracis-19045 142 31 2 2 NUM bracis-19045 142 32 , , PUNCT bracis-19045 142 33 \sigma \sigma PROPN bracis-19045 142 34 _ _ NOUN bracis-19045 142 35 2 2 NUM bracis-19045 142 36 ^ ^ SYM bracis-19045 142 37 2)\ 2)\ NUM bracis-19045 142 38 ) ) PUNCT bracis-19045 142 39 , , PUNCT bracis-19045 142 40 respectively respectively ADV bracis-19045 142 41 . . PUNCT bracis-19045 143 1 then then ADV bracis-19045 143 2 , , PUNCT bracis-19045 143 3 the the DET bracis-19045 143 4 relative relative ADJ bracis-19045 143 5 entropy entropy NOUN bracis-19045 143 6 becomes become VERB bracis-19045 143 7 : : PUNCT bracis-19045 143 8 $ $ SYM bracis-19045 143 9 $ $ SYM bracis-19045 143 10 \begin{aligned \begin{aligne VERB bracis-19045 143 11 } } PUNCT bracis-19045 143 12 d_{kl}(p d_{kl}(p NOUN bracis-19045 143 13 , , PUNCT bracis-19045 143 14 q q X bracis-19045 143 15 ) ) PUNCT bracis-19045 143 16 = = SYM bracis-19045 143 17 log log PROPN bracis-19045 143 18 \left \left PROPN bracis-19045 143 19 ( ( PUNCT bracis-19045 143 20 \frac{\sigma \frac{\sigma X bracis-19045 143 21 _ _ NOUN bracis-19045 143 22 2}{\sigma 2}{\sigma X bracis-19045 144 1 _ _ NOUN bracis-19045 144 2 1 1 X bracis-19045 144 3 } } PUNCT bracis-19045 144 4 \right)&+ \right)&+ VERB bracis-19045 144 5 \frac{1}{2\sigma \frac{1}{2\sigma NOUN bracis-19045 144 6 _ _ PRON bracis-19045 144 7 2 2 NUM bracis-19045 144 8 ^ ^ SYM bracis-19045 144 9 2 2 NUM bracis-19045 144 10 } } PUNCT bracis-19045 144 11 e_p[(x e_p[(x NOUN bracis-19045 144 12 \mu \mu PROPN bracis-19045 144 13 _ _ PROPN bracis-19045 144 14 2)^2 2)^2 NUM bracis-19045 144 15 ] ] PUNCT bracis-19045 144 16 \frac{1}{2\sigma \frac{1}{2\sigma X bracis-19045 145 1 _ _ NOUN bracis-19045 145 2 1 1 NUM bracis-19045 145 3 ^ ^ SYM bracis-19045 145 4 2 2 NUM bracis-19045 145 5 } } PUNCT bracis-19045 145 6 e_p[(x e_p[(x NOUN bracis-19045 145 7 \mu \mu PROPN bracis-19045 145 8 _ _ PROPN bracis-19045 145 9 1)^2 1)^2 NUM bracis-19045 145 10 ] ] PUNCT bracis-19045 145 11 \end{aligned}$$ \end{aligned}$$ X bracis-19045 145 12 ( ( PUNCT bracis-19045 145 13 31 31 NUM bracis-19045 145 14 ) ) PUNCT bracis-19045 145 15 by by ADP bracis-19045 145 16 the the DET bracis-19045 145 17 definition definition NOUN bracis-19045 145 18 of of ADP bracis-19045 145 19 central central ADJ bracis-19045 145 20 moments moment NOUN bracis-19045 145 21 , , PUNCT bracis-19045 145 22 we we PRON bracis-19045 145 23 then then ADV bracis-19045 145 24 have have VERB bracis-19045 145 25 : : PUNCT bracis-19045 145 26 $ $ SYM bracis-19045 145 27 $ $ SYM bracis-19045 145 28 \begin{aligned \begin{aligne VERB bracis-19045 145 29 } } PUNCT bracis-19045 145 30 e_p[(x e_p[(x NOUN bracis-19045 145 31 \mu \mu PROPN bracis-19045 145 32 _ _ PROPN bracis-19045 145 33 1)^2]&= 1)^2]&= PROPN bracis-19045 146 1 \sigma \sigma PROPN bracis-19045 146 2 _ _ NOUN bracis-19045 146 3 1 1 NUM bracis-19045 146 4 ^ ^ SYM bracis-19045 146 5 2 2 NUM bracis-19045 146 6 \end{aligned}$$ \end{aligned}$$ X bracis-19045 146 7 ( ( PUNCT bracis-19045 146 8 32 32 NUM bracis-19045 146 9 ) ) PUNCT bracis-19045 146 10 $ $ SYM bracis-19045 146 11 $ $ SYM bracis-19045 146 12 \begin{aligned \begin{aligne VERB bracis-19045 146 13 } } PUNCT bracis-19045 146 14 e_p[(x e_p[(x NOUN bracis-19045 146 15 \mu \mu PROPN bracis-19045 146 16 _ _ PROPN bracis-19045 146 17 2)^2]&= 2)^2]&= NUM bracis-19045 146 18 e[x^2 e[x^2 X bracis-19045 146 19 ] ] X bracis-19045 146 20 2e[x]\mu 2e[x]\mu NUM bracis-19045 146 21 _ _ SYM bracis-19045 146 22 2 2 NUM bracis-19045 147 1 + + CCONJ bracis-19045 147 2 \mu \mu NOUN bracis-19045 147 3 _ _ NOUN bracis-19045 147 4 2 2 NUM bracis-19045 147 5 ^ ^ SYM bracis-19045 147 6 2 2 NUM bracis-19045 147 7 \end{aligned}$$ \end{aligned}$$ ADP bracis-19045 147 8 ( ( PUNCT bracis-19045 147 9 33 33 NUM bracis-19045 147 10 ) ) PUNCT bracis-19045 147 11 $ $ SYM bracis-19045 147 12 $ $ SYM bracis-19045 147 13 \begin{aligned \begin{aligne VERB bracis-19045 147 14 } } PUNCT bracis-19045 147 15 e[x^2]&= e[x^2]&= PROPN bracis-19045 147 16 var[x var[x PROPN bracis-19045 147 17 ] ] PUNCT bracis-19045 148 1 + + PUNCT bracis-19045 148 2 e^2[x e^2[x SYM bracis-19045 148 3 ] ] X bracis-19045 149 1 = = PUNCT bracis-19045 150 1 \sigma \sigma PROPN bracis-19045 150 2 _ _ NOUN bracis-19045 150 3 1 1 NUM bracis-19045 150 4 ^ ^ SYM bracis-19045 150 5 2 2 NUM bracis-19045 150 6 + + ADP bracis-19045 150 7 \mu \mu NOUN bracis-19045 150 8 _ _ NOUN bracis-19045 150 9 1 1 NUM bracis-19045 150 10 ^ ^ SYM bracis-19045 150 11 2 2 NUM bracis-19045 150 12 \end{aligned}$$ \end{aligned}$$ X bracis-19045 150 13 ( ( PUNCT bracis-19045 150 14 34 34 NUM bracis-19045 150 15 ) ) PUNCT bracis-19045 150 16 which which PRON bracis-19045 150 17 leads lead VERB bracis-19045 150 18 to to ADP bracis-19045 150 19 the the DET bracis-19045 150 20 following follow VERB bracis-19045 150 21 closed closed ADJ bracis-19045 150 22 - - PUNCT bracis-19045 150 23 form form NOUN bracis-19045 150 24 equation equation NOUN bracis-19045 150 25 : : PUNCT bracis-19045 150 26 $ $ SYM bracis-19045 150 27 $ $ SYM bracis-19045 150 28 \begin{aligned \begin{aligne VERB bracis-19045 150 29 } } PUNCT bracis-19045 150 30 d_{kl}(p d_{kl}(p NOUN bracis-19045 150 31 , , PUNCT bracis-19045 150 32 q)&= q)&= X bracis-19045 150 33 log log PROPN bracis-19045 150 34 \left \left PROPN bracis-19045 150 35 ( ( PUNCT bracis-19045 150 36 \frac{\sigma \frac{\sigma X bracis-19045 150 37 _ _ NOUN bracis-19045 150 38 2}{\sigma 2}{\sigma X bracis-19045 151 1 _ _ NOUN bracis-19045 151 2 1 1 X bracis-19045 151 3 } } PUNCT bracis-19045 151 4 \right \right PROPN bracis-19045 151 5 ) ) PUNCT bracis-19045 151 6 + + SYM bracis-19045 151 7 \frac{\sigma \frac{\sigma X bracis-19045 151 8 _ _ NOUN bracis-19045 151 9 1 1 NUM bracis-19045 151 10 ^ ^ SYM bracis-19045 151 11 2 2 NUM bracis-19045 151 12 + + CCONJ bracis-19045 151 13 ( ( PUNCT bracis-19045 151 14 \mu \mu PROPN bracis-19045 151 15 _ _ NOUN bracis-19045 151 16 1 1 NUM bracis-19045 151 17 \mu \mu NOUN bracis-19045 151 18 _ _ PUNCT bracis-19045 151 19 2)^2}{2\sigma 2)^2}{2\sigma X bracis-19045 152 1 _ _ PUNCT bracis-19045 152 2 2 2 NUM bracis-19045 152 3 ^ ^ SYM bracis-19045 152 4 2 2 NUM bracis-19045 152 5 } } PUNCT bracis-19045 152 6 \frac{1}{2 \frac{1}{2 NOUN bracis-19045 152 7 } } PUNCT bracis-19045 152 8 \end{aligned}$$ \end{aligned}$$ X bracis-19045 152 9 ( ( PUNCT bracis-19045 152 10 35 35 NUM bracis-19045 152 11 ) ) PUNCT bracis-19045 152 12 as as SCONJ bracis-19045 152 13 the the DET bracis-19045 152 14 relative relative ADJ bracis-19045 152 15 entropy entropy NOUN bracis-19045 152 16 is be AUX bracis-19045 152 17 not not PART bracis-19045 152 18 symmetric symmetric ADJ bracis-19045 152 19 , , PUNCT bracis-19045 152 20 it it PRON bracis-19045 152 21 is be AUX bracis-19045 152 22 possible possible ADJ bracis-19045 152 23 to to PART bracis-19045 152 24 compute compute VERB bracis-19045 152 25 its its PRON bracis-19045 152 26 symmetrized symmetrize VERB bracis-19045 152 27 counterpart counterpart NOUN bracis-19045 152 28 as as SCONJ bracis-19045 152 29 follows follow VERB bracis-19045 152 30 : : PUNCT bracis-19045 152 31 $ $ SYM bracis-19045 152 32 $ $ SYM bracis-19045 152 33 \begin{aligned}&d_{kl}^{sym}(p \begin{aligned}&d_{kl}^{sym}(p NOUN bracis-19045 152 34 , , PUNCT bracis-19045 152 35 q q X bracis-19045 152 36 ) ) PUNCT bracis-19045 152 37 = = SYM bracis-19045 153 1 \frac{1}{2 \frac{1}{2 NOUN bracis-19045 153 2 } } PUNCT bracis-19045 154 1 [ [ X bracis-19045 154 2 d_{kl}(p d_{kl}(p NOUN bracis-19045 154 3 , , PUNCT bracis-19045 154 4 q q NOUN bracis-19045 154 5 ) ) PUNCT bracis-19045 154 6 + + NUM bracis-19045 154 7 d_{kl}(q d_{kl}(q PROPN bracis-19045 154 8 , , PUNCT bracis-19045 154 9 p p NOUN bracis-19045 154 10 ) ) PUNCT bracis-19045 154 11 ] ] PUNCT bracis-19045 155 1 \\&= \\&= X bracis-19045 155 2 \frac{1}{4 \frac{1}{4 X bracis-19045 155 3 } } PUNCT bracis-19045 155 4 \left \left PROPN bracis-19045 155 5 [ [ PUNCT bracis-19045 155 6 \frac{\sigma \frac{\sigma X bracis-19045 155 7 _ _ NOUN bracis-19045 155 8 1 1 NUM bracis-19045 155 9 ^ ^ SYM bracis-19045 155 10 2 2 NUM bracis-19045 155 11 + + CCONJ bracis-19045 155 12 ( ( PUNCT bracis-19045 155 13 \mu \mu PROPN bracis-19045 155 14 _ _ NOUN bracis-19045 155 15 1 1 NUM bracis-19045 155 16 \mu \mu NOUN bracis-19045 155 17 _ _ NOUN bracis-19045 155 18 2)^2}{\sigma 2)^2}{\sigma NOUN bracis-19045 156 1 _ _ PUNCT bracis-19045 156 2 2 2 NUM bracis-19045 156 3 ^ ^ SYM bracis-19045 156 4 2 2 NUM bracis-19045 156 5 } } PUNCT bracis-19045 156 6 + + NUM bracis-19045 156 7 \frac{\sigma \frac{\sigma NOUN bracis-19045 156 8 _ _ NOUN bracis-19045 156 9 2 2 NUM bracis-19045 156 10 ^ ^ SYM bracis-19045 156 11 2 2 NUM bracis-19045 156 12 + + CCONJ bracis-19045 156 13 ( ( PUNCT bracis-19045 156 14 \mu \mu PROPN bracis-19045 156 15 _ _ NOUN bracis-19045 156 16 1 1 NUM bracis-19045 156 17 \mu \mu NOUN bracis-19045 156 18 _ _ NOUN bracis-19045 156 19 2)^2}{\sigma 2)^2}{\sigma NOUN bracis-19045 156 20 _ _ PUNCT bracis-19045 157 1 1 1 NUM bracis-19045 157 2 ^ ^ SYM bracis-19045 157 3 2 2 NUM bracis-19045 157 4 } } SYM bracis-19045 157 5 2 2 NUM bracis-19045 157 6 \right \right NOUN bracis-19045 157 7 ] ] X bracis-19045 157 8 \nonumber \nonumber PROPN bracis-19045 157 9 \\&= \\&= X bracis-19045 157 10 \frac{1}{4\sigma \frac{1}{4\sigma X bracis-19045 157 11 _ _ NOUN bracis-19045 157 12 1 1 NUM bracis-19045 157 13 ^ ^ SYM bracis-19045 157 14 2\sigma 2\sigma NUM bracis-19045 157 15 _ _ NOUN bracis-19045 157 16 2 2 NUM bracis-19045 157 17 ^ ^ SYM bracis-19045 157 18 2 2 NUM bracis-19045 157 19 } } PUNCT bracis-19045 157 20 \left \left PROPN bracis-19045 157 21 [ [ PUNCT bracis-19045 157 22 \left \left PROPN bracis-19045 157 23 ( ( PUNCT bracis-19045 157 24 \sigma \sigma PROPN bracis-19045 157 25 _ _ NOUN bracis-19045 157 26 1 1 NUM bracis-19045 157 27 ^ ^ SYM bracis-19045 157 28 2 2 NUM bracis-19045 157 29 \sigma \sigma PROPN bracis-19045 157 30 _ _ PROPN bracis-19045 157 31 2 2 NUM bracis-19045 157 32 ^ ^ SYM bracis-19045 157 33 2 2 NUM bracis-19045 157 34 \right \right NOUN bracis-19045 157 35 ) ) PUNCT bracis-19045 157 36 ^2 ^2 PUNCT bracis-19045 158 1 + + X bracis-19045 158 2 \left \left PROPN bracis-19045 158 3 ( ( PUNCT bracis-19045 158 4 \mu \mu PROPN bracis-19045 158 5 _ _ NOUN bracis-19045 158 6 1 1 NUM bracis-19045 158 7 \mu \mu NOUN bracis-19045 158 8 _ _ NOUN bracis-19045 158 9 2 2 NUM bracis-19045 158 10 \right \right NOUN bracis-19045 158 11 ) ) PUNCT bracis-19045 158 12 ^2 ^2 PUNCT bracis-19045 158 13 \left \left PROPN bracis-19045 158 14 ( ( PUNCT bracis-19045 158 15 \sigma \sigma PROPN bracis-19045 158 16 _ _ NOUN bracis-19045 158 17 1 1 NUM bracis-19045 158 18 ^ ^ SYM bracis-19045 158 19 2 2 NUM bracis-19045 158 20 + + NUM bracis-19045 158 21 \sigma \sigma PROPN bracis-19045 158 22 _ _ NOUN bracis-19045 158 23 2 2 NUM bracis-19045 158 24 ^ ^ SYM bracis-19045 158 25 2 2 NUM bracis-19045 158 26 \right \right NOUN bracis-19045 158 27 ) ) PUNCT bracis-19045 158 28 \right \right NOUN bracis-19045 158 29 ] ] X bracis-19045 158 30 \nonumber \nonumber PROPN bracis-19045 158 31 \end{aligned}$$ \end{aligned}$$ PROPN bracis-19045 158 32 ( ( PUNCT bracis-19045 158 33 36 36 NUM bracis-19045 158 34 ) ) PUNCT bracis-19045 158 35 let let VERB bracis-19045 158 36 \(\vec \(\vec NOUN bracis-19045 158 37 { { PUNCT bracis-19045 158 38 d}_{ij}\ d}_{ij}\ NOUN bracis-19045 158 39 ) ) PUNCT bracis-19045 158 40 be be VERB bracis-19045 158 41 the the DET bracis-19045 158 42 m m ADJ bracis-19045 158 43 - - ADJ bracis-19045 158 44 dimensional dimensional ADJ bracis-19045 158 45 vector vector NOUN bracis-19045 158 46 of of ADP bracis-19045 158 47 symmetrized symmetrized ADJ bracis-19045 158 48 relative relative ADJ bracis-19045 158 49 entropies entropy NOUN bracis-19045 158 50 between between ADP bracis-19045 158 51 the the DET bracis-19045 158 52 components component NOUN bracis-19045 158 53 of of ADP bracis-19045 158 54 \(\vec \(\vec X bracis-19045 158 55 { { PUNCT bracis-19045 158 56 p}_i\ p}_i\ NOUN bracis-19045 158 57 ) ) PUNCT bracis-19045 158 58 and and CCONJ bracis-19045 158 59 \(\vec \(\vec NOUN bracis-19045 158 60 { { PUNCT bracis-19045 158 61 p}_j\ p}_j\ NOUN bracis-19045 158 62 ) ) PUNCT bracis-19045 158 63 , , PUNCT bracis-19045 158 64 computed compute VERB bracis-19045 158 65 by by ADP bracis-19045 158 66 the the DET bracis-19045 158 67 direct direct ADJ bracis-19045 158 68 application application NOUN bracis-19045 158 69 of of ADP bracis-19045 158 70 equation equation NOUN bracis-19045 158 71 ( ( PUNCT bracis-19045 158 72 36 36 NUM bracis-19045 158 73 ) ) PUNCT bracis-19045 158 74 in in ADP bracis-19045 158 75 each each DET bracis-19045 158 76 pair pair NOUN bracis-19045 158 77 of of ADP bracis-19045 158 78 parameter parameter NOUN bracis-19045 158 79 vectors vector NOUN bracis-19045 158 80 . . PUNCT bracis-19045 159 1 then then ADV bracis-19045 159 2 , , PUNCT bracis-19045 159 3 in in ADP bracis-19045 159 4 the the DET bracis-19045 159 5 proposed propose VERB bracis-19045 159 6 method method NOUN bracis-19045 159 7 , , PUNCT bracis-19045 159 8 the the DET bracis-19045 159 9 entropic entropic ADJ bracis-19045 159 10 matrix matrix NOUN bracis-19045 159 11 \(c_i\ \(c_i\ NOUN bracis-19045 159 12 ) ) PUNCT bracis-19045 159 13 is be AUX bracis-19045 159 14 computed compute VERB bracis-19045 159 15 as as SCONJ bracis-19045 159 16 follows follow VERB bracis-19045 159 17 : : PUNCT bracis-19045 160 1 $ $ SYM bracis-19045 160 2 $ $ SYM bracis-19045 160 3 \begin{aligned \begin{aligne VERB bracis-19045 160 4 } } PUNCT bracis-19045 160 5 c_i c_i PUNCT bracis-19045 160 6 = = SYM bracis-19045 160 7 \vec \vec PROPN bracis-19045 160 8 { { PUNCT bracis-19045 160 9 d}_{ij}^t d}_{ij}^t PROPN bracis-19045 160 10 \vec \vec PROPN bracis-19045 160 11 { { PUNCT bracis-19045 160 12 d}_{ik d}_{ik PROPN bracis-19045 160 13 } } PUNCT bracis-19045 160 14 \end{aligned}$$ \end{aligned}$$ X bracis-19045 160 15 ( ( PUNCT bracis-19045 160 16 37 37 NUM bracis-19045 160 17 ) ) PUNCT bracis-19045 160 18 where where SCONJ bracis-19045 160 19 the the DET bracis-19045 160 20 column column NOUN bracis-19045 160 21 vector vector NOUN bracis-19045 160 22 \(\vec \(\vec NOUN bracis-19045 160 23 { { PUNCT bracis-19045 160 24 d}_{ij}\ d}_{ij}\ NOUN bracis-19045 160 25 ) ) PUNCT bracis-19045 160 26 involves involve VERB bracis-19045 160 27 the the DET bracis-19045 160 28 patches patch NOUN bracis-19045 160 29 \(\vec \(\vec NOUN bracis-19045 160 30 { { PUNCT bracis-19045 160 31 p}_i\ p}_i\ NOUN bracis-19045 160 32 ) ) PUNCT bracis-19045 160 33 and and CCONJ bracis-19045 160 34 \(\vec \(\vec NOUN bracis-19045 160 35 { { PUNCT bracis-19045 160 36 p}_j\ p}_j\ NOUN bracis-19045 160 37 ) ) PUNCT bracis-19045 160 38 , , PUNCT bracis-19045 160 39 while while SCONJ bracis-19045 160 40 the the DET bracis-19045 160 41 row row NOUN bracis-19045 160 42 vector vector NOUN bracis-19045 160 43 \(\vec \(\vec NOUN bracis-19045 160 44 { { PUNCT bracis-19045 160 45 d}_{ik}^t\ d}_{ik}^t\ NOUN bracis-19045 160 46 ) ) PUNCT bracis-19045 160 47 involves involve VERB bracis-19045 160 48 the the DET bracis-19045 160 49 patches patch NOUN bracis-19045 160 50 \(\vec \(\vec NOUN bracis-19045 160 51 { { PUNCT bracis-19045 160 52 p}_i\ p}_i\ NOUN bracis-19045 160 53 ) ) PUNCT bracis-19045 160 54 and and CCONJ bracis-19045 160 55 \(\vec \(\vec NOUN bracis-19045 160 56 { { PUNCT bracis-19045 160 57 p}_k\ p}_k\ PROPN bracis-19045 160 58 ) ) PUNCT bracis-19045 160 59 . . PUNCT bracis-19045 161 1 it it PRON bracis-19045 161 2 is be AUX bracis-19045 161 3 worth worth ADJ bracis-19045 161 4 noticing notice VERB bracis-19045 161 5 that that SCONJ bracis-19045 161 6 unlike unlike ADP bracis-19045 161 7 the the DET bracis-19045 161 8 established establish VERB bracis-19045 161 9 lle lle PROPN bracis-19045 161 10 method method NOUN bracis-19045 161 11 which which PRON bracis-19045 161 12 employs employ VERB bracis-19045 161 13 the the DET bracis-19045 161 14 pairwise pairwise NOUN bracis-19045 161 15 euclidean euclidean ADJ bracis-19045 161 16 distance distance NOUN bracis-19045 161 17 , , PUNCT bracis-19045 161 18 the the DET bracis-19045 161 19 proposed propose VERB bracis-19045 161 20 lle lle PROPN bracis-19045 161 21 - - PUNCT bracis-19045 161 22 kl kl PROPN bracis-19045 161 23 method method NOUN bracis-19045 161 24 employs employ VERB bracis-19045 161 25 a a DET bracis-19045 161 26 patch patch NOUN bracis-19045 161 27 - - PUNCT bracis-19045 161 28 based base VERB bracis-19045 161 29 distance distance NOUN bracis-19045 161 30 ( ( PUNCT bracis-19045 161 31 relative relative ADJ bracis-19045 161 32 entropy entropy PROPN bracis-19045 161 33 ) ) PUNCT bracis-19045 161 34 instead instead ADV bracis-19045 161 35 , , PUNCT bracis-19045 161 36 becoming become VERB bracis-19045 161 37 less less ADV bracis-19045 161 38 sensitive sensitive ADJ bracis-19045 161 39 to to ADP bracis-19045 161 40 the the DET bracis-19045 161 41 presence presence NOUN bracis-19045 161 42 of of ADP bracis-19045 161 43 noise noise NOUN bracis-19045 161 44 and and CCONJ bracis-19045 161 45 outliers outlier NOUN bracis-19045 161 46 in in ADP bracis-19045 161 47 the the DET bracis-19045 161 48 observed observed ADJ bracis-19045 161 49 data datum NOUN bracis-19045 161 50 . . PUNCT bracis-19045 162 1 the the DET bracis-19045 162 2 remaining remain VERB bracis-19045 162 3 steps step NOUN bracis-19045 162 4 of of ADP bracis-19045 162 5 the the DET bracis-19045 162 6 algorithm algorithm NOUN bracis-19045 162 7 are be AUX bracis-19045 162 8 precisely precisely ADV bracis-19045 162 9 the the DET bracis-19045 162 10 same same ADJ bracis-19045 162 11 as as ADP bracis-19045 162 12 in in ADP bracis-19045 162 13 the the DET bracis-19045 162 14 established establish VERB bracis-19045 162 15 lle lle NOUN bracis-19045 162 16 . . PUNCT bracis-19045 163 1 the the DET bracis-19045 163 2 set set NOUN bracis-19045 163 3 of of ADP bracis-19045 163 4 logical logical ADJ bracis-19045 163 5 steps step NOUN bracis-19045 163 6 comprising comprise VERB bracis-19045 163 7 the the DET bracis-19045 163 8 lle lle PROPN bracis-19045 163 9 - - PUNCT bracis-19045 163 10 kl kl PROPN bracis-19045 163 11 method method NOUN bracis-19045 163 12 is be AUX bracis-19045 163 13 detailed detail VERB bracis-19045 163 14 in in ADP bracis-19045 163 15 algorithm algorithm NOUN bracis-19045 163 16     SPACE bracis-19045 163 17 1 1 NUM bracis-19045 163 18 . . PUNCT bracis-19045 164 1 in in ADP bracis-19045 164 2 the the DET bracis-19045 164 3 subsequent subsequent ADJ bracis-19045 164 4 sections section NOUN bracis-19045 164 5 , , PUNCT bracis-19045 164 6 we we PRON bracis-19045 164 7 present present VERB bracis-19045 164 8 some some DET bracis-19045 164 9 computational computational ADJ bracis-19045 164 10 experiments experiment NOUN bracis-19045 164 11 to to PART bracis-19045 164 12 compare compare VERB bracis-19045 164 13 the the DET bracis-19045 164 14 performance performance NOUN bracis-19045 164 15 of of ADP bracis-19045 164 16 the the DET bracis-19045 164 17 proposed propose VERB bracis-19045 164 18 method method NOUN bracis-19045 164 19 against against ADP bracis-19045 164 20 several several ADJ bracis-19045 164 21 manifold manifold ADJ bracis-19045 164 22 learning learning NOUN bracis-19045 164 23 algorithms algorithm NOUN bracis-19045 164 24 . . PUNCT bracis-19045 165 1 4 4 NUM bracis-19045 165 2 data datum NOUN bracis-19045 165 3 , , PUNCT bracis-19045 165 4 experiments experiment NOUN bracis-19045 165 5 and and CCONJ bracis-19045 165 6 results result NOUN bracis-19045 165 7 in in ADP bracis-19045 165 8 the the DET bracis-19045 165 9 present present ADJ bracis-19045 165 10 section section NOUN bracis-19045 165 11 is be AUX bracis-19045 165 12 presented present VERB bracis-19045 165 13 the the DET bracis-19045 165 14 data datum NOUN bracis-19045 165 15 used use VERB bracis-19045 165 16 , , PUNCT bracis-19045 165 17 experiments experiment NOUN bracis-19045 165 18 performed perform VERB bracis-19045 165 19 , , PUNCT bracis-19045 165 20 and and CCONJ bracis-19045 165 21 a a DET bracis-19045 165 22 discussion discussion NOUN bracis-19045 165 23 on on ADP bracis-19045 165 24 the the DET bracis-19045 165 25 respective respective ADJ bracis-19045 165 26 empirical empirical ADJ bracis-19045 165 27 results result NOUN bracis-19045 165 28 . . PUNCT bracis-19045 166 1 4.1 4.1 NUM bracis-19045 166 2 data datum NOUN bracis-19045 166 3 in in ADP bracis-19045 166 4 the the DET bracis-19045 166 5 present present ADJ bracis-19045 166 6 study study NOUN bracis-19045 166 7 , , PUNCT bracis-19045 166 8 a a DET bracis-19045 166 9 total total NOUN bracis-19045 166 10 of of ADP bracis-19045 166 11 25 25 NUM bracis-19045 166 12 datasets dataset NOUN bracis-19045 166 13 are be AUX bracis-19045 166 14 used use VERB bracis-19045 166 15 as as ADP bracis-19045 166 16 input input NOUN bracis-19045 166 17 data datum NOUN bracis-19045 166 18 . . PUNCT bracis-19045 167 1 such such ADJ bracis-19045 167 2 datasets dataset NOUN bracis-19045 167 3 are be AUX bracis-19045 167 4 widely widely ADV bracis-19045 167 5 used use VERB bracis-19045 167 6 in in ADP bracis-19045 167 7 machine machine NOUN bracis-19045 167 8 learning learning NOUN bracis-19045 167 9 estimations estimation NOUN bracis-19045 167 10 and and CCONJ bracis-19045 167 11 applications application NOUN bracis-19045 167 12 . . PUNCT bracis-19045 168 1 the the DET bracis-19045 168 2 datasets dataset NOUN bracis-19045 168 3 used use VERB bracis-19045 168 4 in in ADP bracis-19045 168 5 the the DET bracis-19045 168 6 present present ADJ bracis-19045 168 7 study study NOUN bracis-19045 168 8 were be AUX bracis-19045 168 9 collected collect VERB bracis-19045 168 10 from from ADP bracis-19045 168 11 openml.org openml.org NOUN bracis-19045 168 12 , , PUNCT bracis-19045 168 13 which which PRON bracis-19045 168 14 contains contain VERB bracis-19045 168 15 detailed detailed ADJ bracis-19045 168 16 information information NOUN bracis-19045 168 17 regarding regard VERB bracis-19045 168 18 the the DET bracis-19045 168 19 number number NOUN bracis-19045 168 20 of of ADP bracis-19045 168 21 instances instance NOUN bracis-19045 168 22 , , PUNCT bracis-19045 168 23 features feature NOUN bracis-19045 168 24 , , PUNCT bracis-19045 168 25 and and CCONJ bracis-19045 168 26 classes class NOUN bracis-19045 168 27 for for ADP bracis-19045 168 28 each each DET bracis-19045 168 29 one one NUM bracis-19045 168 30 of of ADP bracis-19045 168 31 datasets dataset NOUN bracis-19045 168 32 . . PUNCT bracis-19045 169 1 4.2 4.2 NUM bracis-19045 169 2 experiments experiment NOUN bracis-19045 169 3 and and CCONJ bracis-19045 169 4 results result NOUN bracis-19045 169 5 in in ADP bracis-19045 169 6 order order NOUN bracis-19045 169 7 to to PART bracis-19045 169 8 test test VERB bracis-19045 169 9 and and CCONJ bracis-19045 169 10 evaluate evaluate VERB bracis-19045 169 11 the the DET bracis-19045 169 12 proposed propose VERB bracis-19045 169 13 method method NOUN bracis-19045 169 14 , , PUNCT bracis-19045 169 15 a a DET bracis-19045 169 16 series series NOUN bracis-19045 169 17 of of ADP bracis-19045 169 18 computational computational ADJ bracis-19045 169 19 experiments experiment NOUN bracis-19045 169 20 is be AUX bracis-19045 169 21 performed perform VERB bracis-19045 169 22 to to PART bracis-19045 169 23 quantitatively quantitatively ADV bracis-19045 169 24 compare compare VERB bracis-19045 169 25 clustering clustering ADJ bracis-19045 169 26 results result NOUN bracis-19045 169 27 obtained obtain VERB bracis-19045 169 28 subsequently subsequently ADV bracis-19045 169 29 to to ADP bracis-19045 169 30 the the DET bracis-19045 169 31 dimensionality dimensionality NOUN bracis-19045 169 32 reduction reduction NOUN bracis-19045 169 33 into into ADP bracis-19045 169 34 2 2 NUM bracis-19045 169 35 - - PUNCT bracis-19045 169 36 d d NOUN bracis-19045 169 37 spaces space NOUN bracis-19045 169 38 using use VERB bracis-19045 169 39 the the DET bracis-19045 169 40 silhouette silhouette NOUN bracis-19045 169 41 coefficient coefficient NOUN bracis-19045 169 42 ( ( PUNCT bracis-19045 169 43 sc sc NOUN bracis-19045 169 44 ) ) PUNCT bracis-19045 169 45 i.e. i.e. X bracis-19045 169 46 , , PUNCT bracis-19045 169 47 a a DET bracis-19045 169 48 measure measure NOUN bracis-19045 169 49 of of ADP bracis-19045 169 50 goodness goodness NOUN bracis-19045 169 51 - - PUNCT bracis-19045 169 52 of of ADP bracis-19045 169 53 - - PUNCT bracis-19045 169 54 fit fit NOUN bracis-19045 169 55 into into ADP bracis-19045 169 56 a a DET bracis-19045 169 57 low low ADJ bracis-19045 169 58 - - PUNCT bracis-19045 169 59 dimensional dimensional ADJ bracis-19045 169 60 representation representation NOUN bracis-19045 169 61 [ [ X bracis-19045 169 62 8 8 NUM bracis-19045 169 63 ] ] PUNCT bracis-19045 169 64 . . PUNCT bracis-19045 170 1 all all DET bracis-19045 170 2 results result NOUN bracis-19045 170 3 are be AUX bracis-19045 170 4 reported report VERB bracis-19045 170 5 in in ADP bracis-19045 170 6 table table NOUN bracis-19045 170 7     SPACE bracis-19045 170 8 1 1 NUM bracis-19045 170 9 . . PUNCT bracis-19045 171 1 it it PRON bracis-19045 171 2 is be AUX bracis-19045 171 3 worth worth ADJ bracis-19045 171 4 noticing notice VERB bracis-19045 171 5 that that SCONJ bracis-19045 171 6 in in ADP bracis-19045 171 7 23 23 NUM bracis-19045 171 8 out out ADP bracis-19045 171 9 of of ADP bracis-19045 171 10 25 25 NUM bracis-19045 171 11 datasets dataset NOUN bracis-19045 171 12 ( ( PUNCT bracis-19045 171 13 i.e. i.e. X bracis-19045 171 14 , , PUNCT bracis-19045 171 15 92 92 NUM bracis-19045 171 16 % % NOUN bracis-19045 171 17 of of ADP bracis-19045 171 18 the the DET bracis-19045 171 19 cases case NOUN bracis-19045 171 20 ) ) PUNCT bracis-19045 171 21 , , PUNCT bracis-19045 171 22 the the DET bracis-19045 171 23 lle lle PROPN bracis-19045 171 24 - - PUNCT bracis-19045 171 25 kl kl PROPN bracis-19045 171 26 method method NOUN bracis-19045 171 27 obtains obtain VERB bracis-19045 171 28 the the DET bracis-19045 171 29 best good ADJ bracis-19045 171 30 performance performance NOUN bracis-19045 171 31 in in ADP bracis-19045 171 32 terms term NOUN bracis-19045 171 33 of of ADP bracis-19045 171 34 the the DET bracis-19045 171 35 sc sc PROPN bracis-19045 171 36 . . PUNCT bracis-19045 172 1 regarding regard VERB bracis-19045 172 2 the the DET bracis-19045 172 3 means mean NOUN bracis-19045 172 4 and and CCONJ bracis-19045 172 5 medians median NOUN bracis-19045 172 6 , , PUNCT bracis-19045 172 7 one one PRON bracis-19045 172 8 may may AUX bracis-19045 172 9 realize realize VERB bracis-19045 172 10 that that SCONJ bracis-19045 172 11 the the DET bracis-19045 172 12 proposed propose VERB bracis-19045 172 13 method method NOUN bracis-19045 172 14 performs perform VERB bracis-19045 172 15 better well ADV bracis-19045 172 16 in in ADP bracis-19045 172 17 comparison comparison NOUN bracis-19045 172 18 with with ADP bracis-19045 172 19 the the DET bracis-19045 172 20 established establish VERB bracis-19045 172 21 lle lle NOUN bracis-19045 172 22 as as ADV bracis-19045 172 23 well well ADV bracis-19045 172 24 as as ADP bracis-19045 172 25 two two NUM bracis-19045 172 26 of of ADP bracis-19045 172 27 its its PRON bracis-19045 172 28 variations variation NOUN bracis-19045 172 29 ( ( PUNCT bracis-19045 172 30 i.e. i.e. X bracis-19045 172 31 , , PUNCT bracis-19045 172 32 hessian hessian ADJ bracis-19045 172 33 lle lle PROPN bracis-19045 172 34 and and CCONJ bracis-19045 172 35 ltsa ltsa NOUN bracis-19045 172 36 ) ) PUNCT bracis-19045 172 37 . . PUNCT bracis-19045 173 1 in in ADP bracis-19045 173 2 addition addition NOUN bracis-19045 173 3 , , PUNCT bracis-19045 173 4 the the DET bracis-19045 173 5 proposed propose VERB bracis-19045 173 6 method method NOUN bracis-19045 173 7 also also ADV bracis-19045 173 8 achieves achieve VERB bracis-19045 173 9 a a DET bracis-19045 173 10 superior superior ADJ bracis-19045 173 11 performance performance NOUN bracis-19045 173 12 compared compare VERB bracis-19045 173 13 to to ADP bracis-19045 173 14 the the DET bracis-19045 173 15 state state NOUN bracis-19045 173 16 - - PUNCT bracis-19045 173 17 of of ADP bracis-19045 173 18 - - PUNCT bracis-19045 173 19 the the DET bracis-19045 173 20 - - PUNCT bracis-19045 173 21 art art NOUN bracis-19045 173 22 algorithm algorithm NOUN bracis-19045 173 23 umap umap ADJ bracis-19045 173 24 . . PUNCT bracis-19045 174 1 it it PRON bracis-19045 174 2 is be AUX bracis-19045 174 3 worth worth ADJ bracis-19045 174 4 noticing notice VERB bracis-19045 174 5 that that SCONJ bracis-19045 174 6 , , PUNCT bracis-19045 174 7 commonly commonly ADV bracis-19045 174 8 , , PUNCT bracis-19045 174 9 the the DET bracis-19045 174 10 umap umap NOUN bracis-19045 174 11 requires require VERB bracis-19045 174 12 a a DET bracis-19045 174 13 reasonably reasonably ADV bracis-19045 174 14 large large ADJ bracis-19045 174 15 sample sample NOUN bracis-19045 174 16 size size NOUN bracis-19045 174 17 to to PART bracis-19045 174 18 perform perform VERB bracis-19045 174 19 well well ADV bracis-19045 174 20 , , PUNCT bracis-19045 174 21 which which PRON bracis-19045 174 22 is be AUX bracis-19045 174 23 not not PART bracis-19045 174 24 always always ADV bracis-19045 174 25 the the DET bracis-19045 174 26 case case NOUN bracis-19045 174 27 in in ADP bracis-19045 174 28 the the DET bracis-19045 174 29 empirical empirical ADJ bracis-19045 174 30 analysis analysis NOUN bracis-19045 174 31 of of ADP bracis-19045 174 32 the the DET bracis-19045 174 33 present present ADJ bracis-19045 174 34 paper paper NOUN bracis-19045 174 35 . . PUNCT bracis-19045 175 1 table table NOUN bracis-19045 175 2 1 1 NUM bracis-19045 175 3 . . PUNCT bracis-19045 175 4 silhouette silhouette NOUN bracis-19045 175 5 coefficients coefficient NOUN bracis-19045 175 6 ( ( PUNCT bracis-19045 175 7 sc sc NOUN bracis-19045 175 8 ) ) PUNCT bracis-19045 175 9 for for ADP bracis-19045 175 10 clusters cluster NOUN bracis-19045 175 11 produced produce VERB bracis-19045 175 12 by by ADP bracis-19045 175 13 pca pca PROPN bracis-19045 175 14 , , PUNCT bracis-19045 175 15 lle lle PROPN bracis-19045 175 16 , , PUNCT bracis-19045 175 17 isomap isomap NOUN bracis-19045 175 18 , , PUNCT bracis-19045 175 19 hessian hessian ADJ bracis-19045 175 20 lle lle PROPN bracis-19045 175 21 , , PUNCT bracis-19045 175 22 ltsa ltsa NOUN bracis-19045 175 23 , , PUNCT bracis-19045 175 24 umap umap ADJ bracis-19045 175 25 , , PUNCT bracis-19045 175 26 and and CCONJ bracis-19045 175 27 lle lle PROPN bracis-19045 175 28 - - PUNCT bracis-19045 175 29 kl kl PROPN bracis-19045 175 30 , , PUNCT bracis-19045 175 31 based base VERB bracis-19045 175 32 on on ADP bracis-19045 175 33 popular popular ADJ bracis-19045 175 34 machine machine NOUN bracis-19045 175 35 learning learn VERB bracis-19045 175 36 datasets dataset NOUN bracis-19045 175 37 ( ( PUNCT bracis-19045 175 38 2 2 NUM bracis-19045 175 39 - - PUNCT bracis-19045 175 40 d d NOUN bracis-19045 175 41 case).full case).full NOUN bracis-19045 175 42 size size NOUN bracis-19045 175 43 table table NOUN bracis-19045 175 44 to to PART bracis-19045 175 45 check check VERB bracis-19045 175 46 if if SCONJ bracis-19045 175 47 the the DET bracis-19045 175 48 results result NOUN bracis-19045 175 49 obtained obtain VERB bracis-19045 175 50 by by ADP bracis-19045 175 51 the the DET bracis-19045 175 52 proposed propose VERB bracis-19045 175 53 method method NOUN bracis-19045 175 54 are be AUX bracis-19045 175 55 statistically statistically ADV bracis-19045 175 56 superior superior ADJ bracis-19045 175 57 to to ADP bracis-19045 175 58 the the DET bracis-19045 175 59 competing compete VERB bracis-19045 175 60 methods method NOUN bracis-19045 175 61 , , PUNCT bracis-19045 175 62 we we PRON bracis-19045 175 63 perform perform VERB bracis-19045 175 64 a a DET bracis-19045 175 65 friedman friedman NOUN bracis-19045 175 66 test test NOUN bracis-19045 175 67 i.e. i.e. X bracis-19045 175 68 non non ADJ bracis-19045 175 69 - - ADJ bracis-19045 175 70 parametric parametric ADJ bracis-19045 175 71 test test NOUN bracis-19045 175 72 for for ADP bracis-19045 175 73 paired pair VERB bracis-19045 175 74 data datum NOUN bracis-19045 175 75 in in ADP bracis-19045 175 76 case case NOUN bracis-19045 175 77 of of ADP bracis-19045 175 78 more more ADJ bracis-19045 175 79 than than ADP bracis-19045 175 80 two two NUM bracis-19045 175 81 groups group NOUN bracis-19045 175 82 [ [ X bracis-19045 175 83 2 2 NUM bracis-19045 175 84 ] ] PUNCT bracis-19045 175 85 . . PUNCT bracis-19045 176 1 for for ADP bracis-19045 176 2 a a DET bracis-19045 176 3 significant significant ADJ bracis-19045 176 4 level level NOUN bracis-19045 176 5 \(\alpha \(\alpha NOUN bracis-19045 176 6 = = SYM bracis-19045 176 7 0.01\ 0.01\ NUM bracis-19045 176 8 ) ) PUNCT bracis-19045 176 9 , , PUNCT bracis-19045 176 10 we we PRON bracis-19045 176 11 conclude conclude VERB bracis-19045 176 12 that that SCONJ bracis-19045 176 13 there there PRON bracis-19045 176 14 is be VERB bracis-19045 176 15 strong strong ADJ bracis-19045 176 16 evidence evidence NOUN bracis-19045 176 17 to to PART bracis-19045 176 18 reject reject VERB bracis-19045 176 19 the the DET bracis-19045 176 20 null null ADJ bracis-19045 176 21 hypothesis hypothesis NOUN bracis-19045 176 22 that that PRON bracis-19045 176 23 all all DET bracis-19045 176 24 groups group NOUN bracis-19045 176 25 are be AUX bracis-19045 176 26 identical identical ADJ bracis-19045 176 27 ( ( PUNCT bracis-19045 176 28 \(p \(p X bracis-19045 176 29 = = SYM bracis-19045 176 30 1.12 1.12 NUM bracis-19045 176 31 \times \time NOUN bracis-19045 176 32 10^{-11}\ 10^{-11}\ NUM bracis-19045 176 33 ) ) PUNCT bracis-19045 176 34 ) ) PUNCT bracis-19045 176 35 . . PUNCT bracis-19045 177 1 in in ADP bracis-19045 177 2 order order NOUN bracis-19045 177 3 to to PART bracis-19045 177 4 analyze analyze VERB bracis-19045 177 5 which which DET bracis-19045 177 6 groups group NOUN bracis-19045 177 7 are be AUX bracis-19045 177 8 significantly significantly ADV bracis-19045 177 9 different different ADJ bracis-19045 177 10 , , PUNCT bracis-19045 177 11 we we PRON bracis-19045 177 12 then then ADV bracis-19045 177 13 perform perform VERB bracis-19045 177 14 a a DET bracis-19045 177 15 nemenyi nemenyi NOUN bracis-19045 177 16 post post ADJ bracis-19045 177 17 - - ADJ bracis-19045 177 18 hoc hoc ADJ bracis-19045 177 19 test test NOUN bracis-19045 177 20 [ [ X bracis-19045 177 21 3 3 NUM bracis-19045 177 22 ] ] PUNCT bracis-19045 177 23 . . PUNCT bracis-19045 178 1 according accord VERB bracis-19045 178 2 to to ADP bracis-19045 178 3 this this DET bracis-19045 178 4 test test NOUN bracis-19045 178 5 , , PUNCT bracis-19045 178 6 there there PRON bracis-19045 178 7 is be VERB bracis-19045 178 8 strong strong ADJ bracis-19045 178 9 evidence evidence NOUN bracis-19045 178 10 that that SCONJ bracis-19045 178 11 the the DET bracis-19045 178 12 lle lle PROPN bracis-19045 178 13 - - PUNCT bracis-19045 178 14 kl kl PROPN bracis-19045 178 15 method method NOUN bracis-19045 178 16 produces produce VERB bracis-19045 178 17 significantly significantly ADV bracis-19045 178 18 higher high ADJ bracis-19045 178 19 scs scs PROPN bracis-19045 178 20 compared compare VERB bracis-19045 178 21 to to ADP bracis-19045 178 22 the the DET bracis-19045 178 23 pca pca NOUN bracis-19045 178 24 ( ( PUNCT bracis-19045 178 25 \(p \(p X bracis-19045 178 26 < < X bracis-19045 178 27 10^{-3}\ 10^{-3}\ NUM bracis-19045 178 28 ) ) PUNCT bracis-19045 178 29 ) ) PUNCT bracis-19045 178 30 , , PUNCT bracis-19045 178 31 isomap isomap NOUN bracis-19045 178 32 ( ( PUNCT bracis-19045 178 33 \(p \(p X bracis-19045 178 34 < < X bracis-19045 178 35 10^{-3}\ 10^{-3}\ NUM bracis-19045 178 36 ) ) PUNCT bracis-19045 178 37 ) ) PUNCT bracis-19045 178 38 , , PUNCT bracis-19045 178 39 lle lle PROPN bracis-19045 178 40 ( ( PUNCT bracis-19045 178 41 \(p \(p X bracis-19045 178 42 < < X bracis-19045 178 43 10^{-3}\ 10^{-3}\ NUM bracis-19045 178 44 ) ) PUNCT bracis-19045 178 45 ) ) PUNCT bracis-19045 178 46 , , PUNCT bracis-19045 178 47 hessian hessian NOUN bracis-19045 178 48 lle lle PROPN bracis-19045 178 49 ( ( PUNCT bracis-19045 178 50 \(p \(p X bracis-19045 178 51 < < X bracis-19045 178 52 10^{-3}\ 10^{-3}\ NUM bracis-19045 178 53 ) ) PUNCT bracis-19045 178 54 ) ) PUNCT bracis-19045 178 55 , , PUNCT bracis-19045 178 56 ltsa ltsa NOUN bracis-19045 178 57 ( ( PUNCT bracis-19045 178 58 \(p \(p X bracis-19045 178 59 < < X bracis-19045 178 60 10^{-3}\ 10^{-3}\ NUM bracis-19045 178 61 ) ) PUNCT bracis-19045 178 62 ) ) PUNCT bracis-19045 178 63 , , PUNCT bracis-19045 178 64 and and CCONJ bracis-19045 178 65 umap umap ADJ bracis-19045 178 66 ( ( PUNCT bracis-19045 178 67 \(p \(p X bracis-19045 178 68 < < X bracis-19045 178 69 10^{-3}\ 10^{-3}\ NUM bracis-19045 178 70 ) ) PUNCT bracis-19045 178 71 ) ) PUNCT bracis-19045 178 72 . . PUNCT bracis-19045 179 1 the the DET bracis-19045 179 2 proposed propose VERB bracis-19045 179 3 method method NOUN bracis-19045 179 4 also also ADV bracis-19045 179 5 has have VERB bracis-19045 179 6 some some DET bracis-19045 179 7 limitations limitation NOUN bracis-19045 179 8 . . PUNCT bracis-19045 180 1 one one NUM bracis-19045 180 2 caveat caveat NOUN bracis-19045 180 3 of of ADP bracis-19045 180 4 the the DET bracis-19045 180 5 lle lle PROPN bracis-19045 180 6 - - PUNCT bracis-19045 180 7 kl kl PROPN bracis-19045 180 8 which which PRON bracis-19045 180 9 is be AUX bracis-19045 180 10 common common ADJ bracis-19045 180 11 to to ADP bracis-19045 180 12 other other ADJ bracis-19045 180 13 manifold manifold ADJ bracis-19045 180 14 learning learning NOUN bracis-19045 180 15 algorithms algorithm NOUN bracis-19045 180 16 , , PUNCT bracis-19045 180 17 consists consist VERB bracis-19045 180 18 of of ADP bracis-19045 180 19 the the DET bracis-19045 180 20 out out NOUN bracis-19045 180 21 - - PUNCT bracis-19045 180 22 of of ADP bracis-19045 180 23 - - PUNCT bracis-19045 180 24 sample sample NOUN bracis-19045 180 25 problem problem NOUN bracis-19045 180 26 . . PUNCT bracis-19045 181 1 it it PRON bracis-19045 181 2 is be AUX bracis-19045 181 3 not not PART bracis-19045 181 4 clear clear ADJ bracis-19045 181 5 how how SCONJ bracis-19045 181 6 to to PART bracis-19045 181 7 properly properly ADV bracis-19045 181 8 evaluate evaluate VERB bracis-19045 181 9 new new ADJ bracis-19045 181 10 samples sample NOUN bracis-19045 181 11 that that PRON bracis-19045 181 12 are be AUX bracis-19045 181 13 not not PART bracis-19045 181 14 part part NOUN bracis-19045 181 15 of of ADP bracis-19045 181 16 the the DET bracis-19045 181 17 training training NOUN bracis-19045 181 18 set set NOUN bracis-19045 181 19 . . PUNCT bracis-19045 182 1 another another DET bracis-19045 182 2 drawback drawback NOUN bracis-19045 182 3 refers refer VERB bracis-19045 182 4 to to ADP bracis-19045 182 5 the the DET bracis-19045 182 6 definition definition NOUN bracis-19045 182 7 of of ADP bracis-19045 182 8 the the DET bracis-19045 182 9 parameter parameter NOUN bracis-19045 182 10 k k PROPN bracis-19045 182 11 ( ( PUNCT bracis-19045 182 12 i.e. i.e. X bracis-19045 182 13 , , PUNCT bracis-19045 182 14 number number NOUN bracis-19045 182 15 of of ADP bracis-19045 182 16 neighbors neighbor NOUN bracis-19045 182 17 ) ) PUNCT bracis-19045 182 18 that that PRON bracis-19045 182 19 controls control VERB bracis-19045 182 20 the the DET bracis-19045 182 21 patch patch ADJ bracis-19045 182 22 size size NOUN bracis-19045 182 23 . . PUNCT bracis-19045 183 1 previous previous ADJ bracis-19045 183 2 computational computational ADJ bracis-19045 183 3 experiments experiment NOUN bracis-19045 183 4 reveal reveal VERB bracis-19045 183 5 that that SCONJ bracis-19045 183 6 the the DET bracis-19045 183 7 sc sc PROPN bracis-19045 183 8 is be AUX bracis-19045 183 9 rather rather ADV bracis-19045 183 10 sensitive sensitive ADJ bracis-19045 183 11 to to ADP bracis-19045 183 12 changes change NOUN bracis-19045 183 13 in in ADP bracis-19045 183 14 such such DET bracis-19045 183 15 a a DET bracis-19045 183 16 parameter parameter NOUN bracis-19045 183 17 . . PUNCT bracis-19045 184 1 our our PRON bracis-19045 184 2 strategy strategy NOUN bracis-19045 184 3 in in ADP bracis-19045 184 4 this this DET bracis-19045 184 5 case case NOUN bracis-19045 184 6 is be AUX bracis-19045 184 7 then then ADV bracis-19045 184 8 as as SCONJ bracis-19045 184 9 follows follow VERB bracis-19045 184 10 : : PUNCT bracis-19045 184 11 for for ADP bracis-19045 184 12 each each DET bracis-19045 184 13 dataset dataset NOUN bracis-19045 184 14 , , PUNCT bracis-19045 184 15 we we PRON bracis-19045 184 16 build build VERB bracis-19045 184 17 the the DET bracis-19045 184 18 kkn kkn NOUN bracis-19045 184 19 graphs graph NOUN bracis-19045 184 20 for for ADP bracis-19045 184 21 all all DET bracis-19045 184 22 values value NOUN bracis-19045 184 23 of of ADP bracis-19045 184 24 k k PROPN bracis-19045 184 25 in in ADP bracis-19045 184 26 the the DET bracis-19045 184 27 interval interval NOUN bracis-19045 184 28 [ [ X bracis-19045 184 29 2 2 NUM bracis-19045 184 30 , , PUNCT bracis-19045 184 31 40 40 NUM bracis-19045 184 32 ] ] PUNCT bracis-19045 184 33 . . PUNCT bracis-19045 185 1 we we PRON bracis-19045 185 2 select select VERB bracis-19045 185 3 the the DET bracis-19045 185 4 best good ADJ bracis-19045 185 5 model model NOUN bracis-19045 185 6 as as ADP bracis-19045 185 7 the the DET bracis-19045 185 8 one one NOUN bracis-19045 185 9 that that PRON bracis-19045 185 10 maximizes maximize VERB bracis-19045 185 11 the the DET bracis-19045 185 12 sc sc PROPN bracis-19045 185 13 among among ADP bracis-19045 185 14 all all DET bracis-19045 185 15 values value NOUN bracis-19045 185 16 of of ADP bracis-19045 185 17 k. k. NOUN bracis-19045 185 18 although although SCONJ bracis-19045 185 19 we we PRON bracis-19045 185 20 are be AUX bracis-19045 185 21 using use VERB bracis-19045 185 22 the the DET bracis-19045 185 23 class class NOUN bracis-19045 185 24 labels label NOUN bracis-19045 185 25 to to PART bracis-19045 185 26 perform perform VERB bracis-19045 185 27 model model NOUN bracis-19045 185 28 selection selection NOUN bracis-19045 185 29 , , PUNCT bracis-19045 185 30 in in ADP bracis-19045 185 31 fact fact NOUN bracis-19045 185 32 , , PUNCT bracis-19045 185 33 the the DET bracis-19045 185 34 lle lle PROPN bracis-19045 185 35 - - PUNCT bracis-19045 185 36 kl kl PROPN bracis-19045 185 37 method method NOUN bracis-19045 185 38 performs perform VERB bracis-19045 185 39 unsupervised unsupervised ADJ bracis-19045 185 40 metric metric ADJ bracis-19045 185 41 learning learning NOUN bracis-19045 185 42 , , PUNCT bracis-19045 185 43 in in ADP bracis-19045 185 44 the the DET bracis-19045 185 45 sense sense NOUN bracis-19045 185 46 that that SCONJ bracis-19045 185 47 we we PRON bracis-19045 185 48 do do AUX bracis-19045 185 49 not not PART bracis-19045 185 50 employ employ VERB bracis-19045 185 51 the the DET bracis-19045 185 52 class class NOUN bracis-19045 185 53 labels label NOUN bracis-19045 185 54 in in ADP bracis-19045 185 55 the the DET bracis-19045 185 56 dimensionality dimensionality NOUN bracis-19045 185 57 reduction reduction NOUN bracis-19045 185 58 . . PUNCT bracis-19045 186 1 a a DET bracis-19045 186 2 data data NOUN bracis-19045 186 3 visualization visualization NOUN bracis-19045 186 4 comparison comparison NOUN bracis-19045 186 5 of of ADP bracis-19045 186 6 the the DET bracis-19045 186 7 clusters cluster NOUN bracis-19045 186 8 obtained obtain VERB bracis-19045 186 9 through through ADP bracis-19045 186 10 the the DET bracis-19045 186 11 application application NOUN bracis-19045 186 12 of of ADP bracis-19045 186 13 the the DET bracis-19045 186 14 lle lle PROPN bracis-19045 186 15 and and CCONJ bracis-19045 186 16 lle lle PROPN bracis-19045 186 17 - - PUNCT bracis-19045 186 18 kl kl NOUN bracis-19045 186 19 methods method NOUN bracis-19045 186 20 for for ADP bracis-19045 186 21 two two NUM bracis-19045 186 22 distinct distinct ADJ bracis-19045 186 23 datasets dataset NOUN bracis-19045 186 24 ( ( PUNCT bracis-19045 186 25 i.e. i.e. X bracis-19045 186 26 , , PUNCT bracis-19045 186 27 tic tic ADJ bracis-19045 186 28 - - PUNCT bracis-19045 186 29 tac tac NOUN bracis-19045 186 30 - - PUNCT bracis-19045 186 31 toe toe NOUN bracis-19045 186 32 and and CCONJ bracis-19045 186 33 corral corral NOUN bracis-19045 186 34 ) ) PUNCT bracis-19045 186 35 are be AUX bracis-19045 186 36 depicted depict VERB bracis-19045 186 37 in in ADP bracis-19045 186 38 figs fig NOUN bracis-19045 186 39 . . PUNCT bracis-19045 186 40     SPACE bracis-19045 186 41 1 1 NUM bracis-19045 186 42 and and CCONJ bracis-19045 186 43 2 2 NUM bracis-19045 186 44 . . X bracis-19045 187 1 it it PRON bracis-19045 187 2 is be AUX bracis-19045 187 3 worth worth ADJ bracis-19045 187 4 noticing notice VERB bracis-19045 187 5 that that SCONJ bracis-19045 187 6 the the DET bracis-19045 187 7 discrimination discrimination NOUN bracis-19045 187 8 between between ADP bracis-19045 187 9 the the DET bracis-19045 187 10 two two NUM bracis-19045 187 11 classes class NOUN bracis-19045 187 12 is be AUX bracis-19045 187 13 more more ADV bracis-19045 187 14 evident evident ADJ bracis-19045 187 15 in in ADP bracis-19045 187 16 the the DET bracis-19045 187 17 proposed propose VERB bracis-19045 187 18 method method NOUN bracis-19045 187 19 than than ADP bracis-19045 187 20 in in ADP bracis-19045 187 21 the the DET bracis-19045 187 22 established establish VERB bracis-19045 187 23 version version NOUN bracis-19045 187 24 of of ADP bracis-19045 187 25 the the DET bracis-19045 187 26 lle lle NOUN bracis-19045 187 27 , , PUNCT bracis-19045 187 28 since since SCONJ bracis-19045 187 29 there there PRON bracis-19045 187 30 is be VERB bracis-19045 187 31 visibly visibly ADV bracis-19045 187 32 less less ADJ bracis-19045 187 33 overlap overlap NOUN bracis-19045 187 34 between between ADP bracis-19045 187 35 the the DET bracis-19045 187 36 blue blue ADJ bracis-19045 187 37 and and CCONJ bracis-19045 187 38 red red ADJ bracis-19045 187 39 cluster cluster NOUN bracis-19045 187 40 . . PUNCT bracis-19045 188 1 fig fig NOUN bracis-19045 188 2 . . PUNCT bracis-19045 189 1 1 1 X bracis-19045 189 2 . . X bracis-19045 189 3 comparison comparison NOUN bracis-19045 189 4 between between ADP bracis-19045 189 5 clusters cluster NOUN bracis-19045 189 6 generated generate VERB bracis-19045 189 7 subsequently subsequently ADV bracis-19045 189 8 to to ADP bracis-19045 189 9 the the DET bracis-19045 189 10 application application NOUN bracis-19045 189 11 of of ADP bracis-19045 189 12 the the DET bracis-19045 189 13 lle lle NOUN bracis-19045 189 14 , , PUNCT bracis-19045 189 15 lle lle PROPN bracis-19045 189 16 - - PUNCT bracis-19045 189 17 kl kl PROPN bracis-19045 189 18 ( ( PUNCT bracis-19045 189 19 with with ADP bracis-19045 189 20 the the DET bracis-19045 189 21 number number NOUN bracis-19045 189 22 of of ADP bracis-19045 189 23 neighbors neighbor NOUN bracis-19045 189 24 set set VERB bracis-19045 189 25 as as ADP bracis-19045 189 26 k k PROPN bracis-19045 189 27 = = NOUN bracis-19045 189 28 9 9 NUM bracis-19045 189 29 ) ) PUNCT bracis-19045 189 30 , , PUNCT bracis-19045 189 31 and and CCONJ bracis-19045 189 32 umap umap VERB bracis-19045 189 33 for for ADP bracis-19045 189 34 the the DET bracis-19045 189 35 tic tic ADJ bracis-19045 189 36 - - PUNCT bracis-19045 189 37 tac tac NOUN bracis-19045 189 38 - - PUNCT bracis-19045 189 39 toe toe NOUN bracis-19045 189 40 dataset dataset NOUN bracis-19045 189 41 . . PUNCT bracis-19045 190 1 full full ADJ bracis-19045 190 2 size size NOUN bracis-19045 190 3 image image NOUN bracis-19045 190 4 fig fig NOUN bracis-19045 190 5 . . PUNCT bracis-19045 191 1 2 2 X bracis-19045 191 2 . . X bracis-19045 191 3 comparison comparison NOUN bracis-19045 191 4 between between ADP bracis-19045 191 5 clusters cluster NOUN bracis-19045 191 6 generated generate VERB bracis-19045 191 7 subsequently subsequently ADV bracis-19045 191 8 to to ADP bracis-19045 191 9 the the DET bracis-19045 191 10 application application NOUN bracis-19045 191 11 of of ADP bracis-19045 191 12 the the DET bracis-19045 191 13 lle lle NOUN bracis-19045 191 14 , , PUNCT bracis-19045 191 15 lle lle PROPN bracis-19045 191 16 - - PUNCT bracis-19045 191 17 kl kl PROPN bracis-19045 191 18 ( ( PUNCT bracis-19045 191 19 with with ADP bracis-19045 191 20 the the DET bracis-19045 191 21 number number NOUN bracis-19045 191 22 of of ADP bracis-19045 191 23 neighbors neighbor NOUN bracis-19045 191 24 set set VERB bracis-19045 191 25 as as ADP bracis-19045 191 26 k k PROPN bracis-19045 191 27 = = NOUN bracis-19045 191 28 14 14 NUM bracis-19045 191 29 ) ) PUNCT bracis-19045 191 30 , , PUNCT bracis-19045 191 31 and and CCONJ bracis-19045 191 32 umap umap VERB bracis-19045 191 33 for for ADP bracis-19045 191 34 the the DET bracis-19045 191 35 corral corral ADJ bracis-19045 191 36 dataset dataset NOUN bracis-19045 191 37 . . PUNCT bracis-19045 192 1 full full ADJ bracis-19045 192 2 size size NOUN bracis-19045 192 3 image image NOUN bracis-19045 192 4 5 5 NUM bracis-19045 192 5 conclusion conclusion NOUN bracis-19045 192 6 in in ADP bracis-19045 192 7 the the DET bracis-19045 192 8 present present ADJ bracis-19045 192 9 study study NOUN bracis-19045 192 10 , , PUNCT bracis-19045 192 11 an an DET bracis-19045 192 12 information information NOUN bracis-19045 192 13 - - PUNCT bracis-19045 192 14 theoretic theoretic NOUN bracis-19045 192 15 lle lle NOUN bracis-19045 192 16 is be AUX bracis-19045 192 17 introduced introduce VERB bracis-19045 192 18 to to PART bracis-19045 192 19 incorporate incorporate VERB bracis-19045 192 20 the the DET bracis-19045 192 21 kl kl PROPN bracis-19045 192 22 divergence divergence NOUN bracis-19045 192 23 between between ADP bracis-19045 192 24 local local ADJ bracis-19045 192 25 gaussian gaussian ADJ bracis-19045 192 26 distributions distribution NOUN bracis-19045 192 27 into into ADP bracis-19045 192 28 the the DET bracis-19045 192 29 knn knn PROPN bracis-19045 192 30 adjacency adjacency PROPN bracis-19045 192 31 graph graph NOUN bracis-19045 192 32 . . PUNCT bracis-19045 193 1 the the DET bracis-19045 193 2 rationale rationale NOUN bracis-19045 193 3 of of ADP bracis-19045 193 4 the the DET bracis-19045 193 5 proposed propose VERB bracis-19045 193 6 method method NOUN bracis-19045 193 7 is be AUX bracis-19045 193 8 to to PART bracis-19045 193 9 replace replace VERB bracis-19045 193 10 the the DET bracis-19045 193 11 pointwise pointwise ADJ bracis-19045 193 12 euclidean euclidean ADJ bracis-19045 193 13 distance distance NOUN bracis-19045 193 14 by by ADP bracis-19045 193 15 a a DET bracis-19045 193 16 more more ADV bracis-19045 193 17 appropriate appropriate ADJ bracis-19045 193 18 patch patch NOUN bracis-19045 193 19 - - PUNCT bracis-19045 193 20 based base VERB bracis-19045 193 21 distance distance NOUN bracis-19045 193 22 . . PUNCT bracis-19045 194 1 such such DET bracis-19045 194 2 a a DET bracis-19045 194 3 methodological methodological ADJ bracis-19045 194 4 modification modification NOUN bracis-19045 194 5 results result NOUN bracis-19045 194 6 in in ADP bracis-19045 194 7 a a DET bracis-19045 194 8 more more ADV bracis-19045 194 9 robust robust ADJ bracis-19045 194 10 method method NOUN bracis-19045 194 11 against against ADP bracis-19045 194 12 the the DET bracis-19045 194 13 presence presence NOUN bracis-19045 194 14 of of ADP bracis-19045 194 15 noise noise NOUN bracis-19045 194 16 or or CCONJ bracis-19045 194 17 outliers outlier NOUN bracis-19045 194 18 in in ADP bracis-19045 194 19 the the DET bracis-19045 194 20 data datum NOUN bracis-19045 194 21 . . PUNCT bracis-19045 195 1 our our PRON bracis-19045 195 2 claim claim NOUN bracis-19045 195 3 is be AUX bracis-19045 195 4 that that SCONJ bracis-19045 195 5 the the DET bracis-19045 195 6 proposed propose VERB bracis-19045 195 7 lle lle PROPN bracis-19045 195 8 - - PUNCT bracis-19045 195 9 kl kl PROPN bracis-19045 195 10 is be AUX bracis-19045 195 11 a a DET bracis-19045 195 12 promising promising ADJ bracis-19045 195 13 alternative alternative NOUN bracis-19045 195 14 to to ADP bracis-19045 195 15 the the DET bracis-19045 195 16 existing exist VERB bracis-19045 195 17 manifold manifold ADJ bracis-19045 195 18 learning learning NOUN bracis-19045 195 19 algorithms algorithm NOUN bracis-19045 195 20 due due ADP bracis-19045 195 21 to to ADP bracis-19045 195 22 the the DET bracis-19045 195 23 proposition proposition NOUN bracis-19045 195 24 of of ADP bracis-19045 195 25 its its PRON bracis-19045 195 26 theoretical theoretical ADJ bracis-19045 195 27 methodological methodological ADJ bracis-19045 195 28 improvements improvement NOUN bracis-19045 195 29 and and CCONJ bracis-19045 195 30 considering consider VERB bracis-19045 195 31 our our PRON bracis-19045 195 32 computational computational ADJ bracis-19045 195 33 experiments experiment NOUN bracis-19045 195 34 that that PRON bracis-19045 195 35 support support VERB bracis-19045 195 36 two two NUM bracis-19045 195 37 main main ADJ bracis-19045 195 38 points point NOUN bracis-19045 195 39 . . PUNCT bracis-19045 196 1 firstly firstly ADV bracis-19045 196 2 , , PUNCT bracis-19045 196 3 the the DET bracis-19045 196 4 quality quality NOUN bracis-19045 196 5 of of ADP bracis-19045 196 6 the the DET bracis-19045 196 7 clusters cluster NOUN bracis-19045 196 8 produced produce VERB bracis-19045 196 9 by by ADP bracis-19045 196 10 the the DET bracis-19045 196 11 lle lle PROPN bracis-19045 196 12 - - PUNCT bracis-19045 196 13 kl kl PROPN bracis-19045 196 14 method method NOUN bracis-19045 196 15 indicates indicate VERB bracis-19045 196 16 a a DET bracis-19045 196 17 superior superior ADJ bracis-19045 196 18 performance performance NOUN bracis-19045 196 19 than than ADP bracis-19045 196 20 those those PRON bracis-19045 196 21 obtained obtain VERB bracis-19045 196 22 by by ADP bracis-19045 196 23 competing compete VERB bracis-19045 196 24 state state NOUN bracis-19045 196 25 - - PUNCT bracis-19045 196 26 of of ADP bracis-19045 196 27 - - PUNCT bracis-19045 196 28 the the DET bracis-19045 196 29 - - PUNCT bracis-19045 196 30 art art NOUN bracis-19045 196 31 manifold manifold ADJ bracis-19045 196 32 learning learning NOUN bracis-19045 196 33 algorithms algorithm NOUN bracis-19045 196 34 . . PUNCT bracis-19045 197 1 secondly secondly ADV bracis-19045 197 2 , , PUNCT bracis-19045 197 3 the the DET bracis-19045 197 4 lle lle PROPN bracis-19045 197 5 - - PUNCT bracis-19045 197 6 kl kl NOUN bracis-19045 197 7 non non ADJ bracis-19045 197 8 - - ADJ bracis-19045 197 9 linear linear ADJ bracis-19045 197 10 features feature NOUN bracis-19045 197 11 may may AUX bracis-19045 197 12 be be AUX bracis-19045 197 13 more more ADJ bracis-19045 197 14 discriminative discriminative NOUN bracis-19045 197 15 in in ADP bracis-19045 197 16 supervised supervised ADJ bracis-19045 197 17 classification classification NOUN bracis-19045 197 18 than than ADP bracis-19045 197 19 features feature NOUN bracis-19045 197 20 obtained obtain VERB bracis-19045 197 21 by by ADP bracis-19045 197 22 state state NOUN bracis-19045 197 23 - - PUNCT bracis-19045 197 24 of of ADP bracis-19045 197 25 - - PUNCT bracis-19045 197 26 the the DET bracis-19045 197 27 - - PUNCT bracis-19045 197 28 art art NOUN bracis-19045 197 29 manifold manifold ADJ bracis-19045 197 30 learning learning NOUN bracis-19045 197 31 algorithms algorithm NOUN bracis-19045 197 32 . . PUNCT bracis-19045 198 1 suggestions suggestion NOUN bracis-19045 198 2 of of ADP bracis-19045 198 3 future future ADJ bracis-19045 198 4 work work NOUN bracis-19045 198 5 include include VERB bracis-19045 198 6 the the DET bracis-19045 198 7 use use NOUN bracis-19045 198 8 of of ADP bracis-19045 198 9 other other ADJ bracis-19045 198 10 information information NOUN bracis-19045 198 11 - - PUNCT bracis-19045 198 12 theoretic theoretic NOUN bracis-19045 198 13 distances distance NOUN bracis-19045 198 14 , , PUNCT bracis-19045 198 15 such such ADJ bracis-19045 198 16 as as ADP bracis-19045 198 17 the the DET bracis-19045 198 18 bhattacharyya bhattacharyya ADJ bracis-19045 198 19 and and CCONJ bracis-19045 198 20 hellinger hellinger ADJ bracis-19045 198 21 distances distance NOUN bracis-19045 198 22 , , PUNCT bracis-19045 198 23 as as ADV bracis-19045 198 24 well well ADV bracis-19045 198 25 as as ADP bracis-19045 198 26 geodesic geodesic ADJ bracis-19045 198 27 distances distance NOUN bracis-19045 198 28 based base VERB bracis-19045 198 29 on on ADP bracis-19045 198 30 the the DET bracis-19045 198 31 fisher fisher PROPN bracis-19045 198 32 information information NOUN bracis-19045 198 33 matrix matrix NOUN bracis-19045 198 34 . . PUNCT bracis-19045 199 1 another another DET bracis-19045 199 2 possibility possibility NOUN bracis-19045 199 3 is be AUX bracis-19045 199 4 the the DET bracis-19045 199 5 non non ADJ bracis-19045 199 6 - - ADJ bracis-19045 199 7 parametric parametric ADJ bracis-19045 199 8 estimation estimation NOUN bracis-19045 199 9 of of ADP bracis-19045 199 10 the the DET bracis-19045 199 11 local local ADJ bracis-19045 199 12 densities density NOUN bracis-19045 199 13 using use VERB bracis-19045 199 14 kernel kernel PROPN bracis-19045 199 15 density density PROPN bracis-19045 199 16 estimation estimation NOUN bracis-19045 199 17 techniques technique NOUN bracis-19045 199 18 ( ( PUNCT bracis-19045 199 19 kde kde NOUN bracis-19045 199 20 ) ) PUNCT bracis-19045 199 21 . . PUNCT bracis-19045 200 1 in in ADP bracis-19045 200 2 this this DET bracis-19045 200 3 case case NOUN bracis-19045 200 4 , , PUNCT bracis-19045 200 5 non non ADJ bracis-19045 200 6 - - ADJ bracis-19045 200 7 parametric parametric ADJ bracis-19045 200 8 versions version NOUN bracis-19045 200 9 of of ADP bracis-19045 200 10 the the DET bracis-19045 200 11 information information NOUN bracis-19045 200 12 - - PUNCT bracis-19045 200 13 theoretic theoretic NOUN bracis-19045 200 14 distances distance NOUN bracis-19045 200 15 may may AUX bracis-19045 200 16 be be AUX bracis-19045 200 17 employed employ VERB bracis-19045 200 18 to to PART bracis-19045 200 19 compute compute VERB bracis-19045 200 20 a a DET bracis-19045 200 21 distance distance NOUN bracis-19045 200 22 function function NOUN bracis-19045 200 23 between between ADP bracis-19045 200 24 the the DET bracis-19045 200 25 patches patch NOUN bracis-19045 200 26 of of ADP bracis-19045 200 27 the the DET bracis-19045 200 28 knn knn NOUN bracis-19045 200 29 graph graph NOUN bracis-19045 200 30 . . PUNCT bracis-19045 201 1 the the DET bracis-19045 201 2 \(\epsilon \(\epsilon NOUN bracis-19045 201 3 \)-neighborhood \)-neighborhood NOUN bracis-19045 201 4 rule rule NOUN bracis-19045 201 5 may may AUX bracis-19045 201 6 also also ADV bracis-19045 201 7 be be AUX bracis-19045 201 8 used use VERB bracis-19045 201 9 for for ADP bracis-19045 201 10 building build VERB bracis-19045 201 11 the the DET bracis-19045 201 12 adjacency adjacency NOUN bracis-19045 201 13 relations relation NOUN bracis-19045 201 14 that that PRON bracis-19045 201 15 define define VERB bracis-19045 201 16 the the DET bracis-19045 201 17 discrete discrete ADJ bracis-19045 201 18 approximation approximation NOUN bracis-19045 201 19 for for ADP bracis-19045 201 20 the the DET bracis-19045 201 21 manifold manifold NOUN bracis-19045 201 22 , , PUNCT bracis-19045 201 23 leading lead VERB bracis-19045 201 24 to to ADP bracis-19045 201 25 non non ADJ bracis-19045 201 26 - - ADJ bracis-19045 201 27 regular regular ADJ bracis-19045 201 28 graphs graph NOUN bracis-19045 201 29 . . PUNCT bracis-19045 202 1 furthermore furthermore ADV bracis-19045 202 2 , , PUNCT bracis-19045 202 3 a a DET bracis-19045 202 4 supervised supervised ADJ bracis-19045 202 5 dimensionality dimensionality NOUN bracis-19045 202 6 reduction reduction NOUN bracis-19045 202 7 based base VERB bracis-19045 202 8 metric metric ADJ bracis-19045 202 9 learning learning NOUN bracis-19045 202 10 approach approach NOUN bracis-19045 202 11 may may AUX bracis-19045 202 12 be be AUX bracis-19045 202 13 created create VERB bracis-19045 202 14 by by ADP bracis-19045 202 15 removing remove VERB bracis-19045 202 16 the the DET bracis-19045 202 17 edges edge NOUN bracis-19045 202 18 of of ADP bracis-19045 202 19 the the DET bracis-19045 202 20 knn knn NOUN bracis-19045 202 21 graph graph NOUN bracis-19045 202 22 for for ADP bracis-19045 202 23 which which PRON bracis-19045 202 24 the the DET bracis-19045 202 25 endpoints endpoint NOUN bracis-19045 202 26 belong belong VERB bracis-19045 202 27 to to ADP bracis-19045 202 28 different different ADJ bracis-19045 202 29 classes class NOUN bracis-19045 202 30 as as ADP bracis-19045 202 31 a a DET bracis-19045 202 32 manner manner NOUN bracis-19045 202 33 to to PART bracis-19045 202 34 impose impose VERB bracis-19045 202 35 that that SCONJ bracis-19045 202 36 the the DET bracis-19045 202 37 optimal optimal ADJ bracis-19045 202 38 reconstruction reconstruction NOUN bracis-19045 202 39 weights weight NOUN bracis-19045 202 40 use use VERB bracis-19045 202 41 only only ADV bracis-19045 202 42 the the DET bracis-19045 202 43 neighbors neighbor NOUN bracis-19045 202 44 that that PRON bracis-19045 202 45 belong belong VERB bracis-19045 202 46 to to ADP bracis-19045 202 47 the the DET bracis-19045 202 48 same same ADJ bracis-19045 202 49 class class NOUN bracis-19045 202 50 within within ADP bracis-19045 202 51 the the DET bracis-19045 202 52 sample sample NOUN bracis-19045 202 53 . . PUNCT bracis-19045 203 1 references reference NOUN bracis-19045 203 2 donoho donoho PROPN bracis-19045 203 3 , , PUNCT bracis-19045 203 4 d.l d.l PROPN bracis-19045 203 5 . . PROPN bracis-19045 203 6 , , PUNCT bracis-19045 203 7 grimes grimes PROPN bracis-19045 203 8 , , PUNCT bracis-19045 203 9 c. c. NOUN bracis-19045 203 10 : : PUNCT bracis-19045 203 11 hessian hessian NOUN bracis-19045 203 12 eigenmaps eigenmap NOUN bracis-19045 203 13 : : PUNCT bracis-19045 203 14 locally locally ADV bracis-19045 203 15 linear linear VERB bracis-19045 203 16 embedding embed VERB bracis-19045 203 17 techniques technique NOUN bracis-19045 203 18 for for ADP bracis-19045 203 19 high high ADJ bracis-19045 203 20 - - PUNCT bracis-19045 203 21 dimensional dimensional ADJ bracis-19045 203 22 data datum NOUN bracis-19045 203 23 . . PUNCT bracis-19045 204 1 proc proc PROPN bracis-19045 204 2 . . PUNCT bracis-19045 205 1 natl natl PROPN bracis-19045 205 2 . . PUNCT bracis-19045 206 1 acad acad PROPN bracis-19045 206 2 . . PUNCT bracis-19045 207 1 sci sci PROPN bracis-19045 207 2 . . PROPN bracis-19045 207 3 100(10 100(10 NUM bracis-19045 207 4 ) ) PUNCT bracis-19045 207 5 , , PUNCT bracis-19045 207 6 5591–5596 5591–5596 NUM bracis-19045 207 7 ( ( PUNCT bracis-19045 207 8 2003 2003 NUM bracis-19045 207 9 ) ) PUNCT bracis-19045 207 10 article article NOUN bracis-19045 207 11     SPACE bracis-19045 207 12 mathscinet mathscinet NOUN bracis-19045 207 13     SPACE bracis-19045 207 14 google google PROPN bracis-19045 207 15 scholar scholar NOUN bracis-19045 207 16     SPACE bracis-19045 207 17 friedman friedman NOUN bracis-19045 207 18 , , PUNCT bracis-19045 207 19 m. m. NOUN bracis-19045 207 20 : : PUNCT bracis-19045 207 21 the the DET bracis-19045 207 22 use use NOUN bracis-19045 207 23 of of ADP bracis-19045 207 24 ranks rank NOUN bracis-19045 207 25 to to PART bracis-19045 207 26 avoid avoid VERB bracis-19045 207 27 the the DET bracis-19045 207 28 assumption assumption NOUN bracis-19045 207 29 of of ADP bracis-19045 207 30 normality normality NOUN bracis-19045 207 31 implicit implicit NOUN bracis-19045 207 32 in in ADP bracis-19045 207 33 the the DET bracis-19045 207 34 analysis analysis NOUN bracis-19045 207 35 of of ADP bracis-19045 207 36 variance variance NOUN bracis-19045 207 37 . . PUNCT bracis-19045 208 1 j. j. PROPN bracis-19045 208 2 am am PROPN bracis-19045 208 3 . . PUNCT bracis-19045 209 1 stat stat PROPN bracis-19045 209 2 . . PUNCT bracis-19045 210 1 assoc assoc PROPN bracis-19045 210 2 . . PUNCT bracis-19045 211 1 32(200 32(200 PROPN bracis-19045 211 2 ) ) PUNCT bracis-19045 211 3 , , PUNCT bracis-19045 211 4 675–701 675–701 NUM bracis-19045 211 5 ( ( PUNCT bracis-19045 211 6 1937 1937 NUM bracis-19045 211 7 ) ) PUNCT bracis-19045 211 8 article article NOUN bracis-19045 211 9     SPACE bracis-19045 211 10 google google PROPN bracis-19045 211 11 scholar scholar NOUN bracis-19045 211 12     SPACE bracis-19045 211 13 hollander hollander NOUN bracis-19045 211 14 , , PUNCT bracis-19045 211 15 m. m. NOUN bracis-19045 211 16 , , PUNCT bracis-19045 211 17 wolfe wolfe PROPN bracis-19045 211 18 , , PUNCT bracis-19045 211 19 d.a d.a PROPN bracis-19045 211 20 . . PROPN bracis-19045 211 21 , , PUNCT bracis-19045 211 22 chicken chicken PROPN bracis-19045 211 23 , , PUNCT bracis-19045 211 24 e. e. PROPN bracis-19045 211 25 : : PUNCT bracis-19045 211 26 nonparametric nonparametric ADJ bracis-19045 211 27 statistical statistical ADJ bracis-19045 211 28 methods method NOUN bracis-19045 211 29 , , PUNCT bracis-19045 211 30 3 3 NUM bracis-19045 211 31 edn edn NOUN bracis-19045 211 32 . . PUNCT bracis-19045 212 1 wiley wiley PROPN bracis-19045 212 2 , , PUNCT bracis-19045 212 3 new new PROPN bracis-19045 212 4 york york PROPN bracis-19045 212 5 ( ( PUNCT bracis-19045 212 6 2015 2015 NUM bracis-19045 212 7 ) ) PUNCT bracis-19045 212 8 google google NOUN bracis-19045 212 9 scholar scholar NOUN bracis-19045 212 10     SPACE bracis-19045 212 11 kullback kullback NOUN bracis-19045 212 12 , , PUNCT bracis-19045 212 13 s. s. PROPN bracis-19045 212 14 , , PUNCT bracis-19045 212 15 leibler leibler PROPN bracis-19045 212 16 , , PUNCT bracis-19045 212 17 r.a r.a PROPN bracis-19045 212 18 . . PROPN bracis-19045 212 19 : : PUNCT bracis-19045 213 1 on on ADP bracis-19045 213 2 information information NOUN bracis-19045 213 3 and and CCONJ bracis-19045 213 4 sufficiency sufficiency NOUN bracis-19045 213 5 . . PUNCT bracis-19045 214 1 ann ann PROPN bracis-19045 214 2 . . PUNCT bracis-19045 214 3 math math PROPN bracis-19045 214 4 . . PUNCT bracis-19045 215 1 stat stat PROPN bracis-19045 215 2 . . PUNCT bracis-19045 216 1 22(1 22(1 NUM bracis-19045 216 2 ) ) PUNCT bracis-19045 217 1 , , PUNCT bracis-19045 217 2 79–86 79–86 NUM bracis-19045 217 3 ( ( PUNCT bracis-19045 217 4 1951 1951 NUM bracis-19045 217 5 ) ) PUNCT bracis-19045 217 6 article article NOUN bracis-19045 217 7     SPACE bracis-19045 217 8 mathscinet mathscinet NOUN bracis-19045 217 9     SPACE bracis-19045 217 10 google google PROPN bracis-19045 217 11 scholar scholar NOUN bracis-19045 217 12     SPACE bracis-19045 217 13 levada levada PROPN bracis-19045 217 14 , , PUNCT bracis-19045 217 15 a.l.m a.l.m NOUN bracis-19045 217 16 . . PUNCT bracis-19045 217 17 : : PUNCT bracis-19045 217 18 parametric parametric ADJ bracis-19045 217 19 pca pca PROPN bracis-19045 217 20 for for ADP bracis-19045 217 21 unsupervised unsupervised ADJ bracis-19045 217 22 metric metric ADJ bracis-19045 217 23 learning learning NOUN bracis-19045 217 24 . . PUNCT bracis-19045 218 1 pattern pattern PROPN bracis-19045 218 2 recognit recognit VERB bracis-19045 218 3 . . PUNCT bracis-19045 219 1 lett lett PROPN bracis-19045 219 2 . . PUNCT bracis-19045 220 1 135 135 NUM bracis-19045 220 2 , , PUNCT bracis-19045 220 3 425–430 425–430 NUM bracis-19045 220 4 ( ( PUNCT bracis-19045 220 5 2020 2020 NUM bracis-19045 220 6 ) ) PUNCT bracis-19045 220 7 article article NOUN bracis-19045 220 8     SPACE bracis-19045 220 9 google google PROPN bracis-19045 220 10 scholar scholar NOUN bracis-19045 220 11     SPACE bracis-19045 220 12 mcinnes mcinne NOUN bracis-19045 220 13 , , PUNCT bracis-19045 220 14 l. l. PROPN bracis-19045 220 15 , , PUNCT bracis-19045 220 16 healy healy PROPN bracis-19045 220 17 , , PUNCT bracis-19045 220 18 j. j. PROPN bracis-19045 220 19 , , PUNCT bracis-19045 220 20 melville melville PROPN bracis-19045 220 21 , , PUNCT bracis-19045 220 22 j. j. PROPN bracis-19045 220 23 : : PUNCT bracis-19045 220 24 umap umap ADJ bracis-19045 220 25 : : PUNCT bracis-19045 220 26 uniform uniform ADJ bracis-19045 220 27 manifold manifold ADJ bracis-19045 220 28 approximation approximation NOUN bracis-19045 220 29 and and CCONJ bracis-19045 220 30 projection projection NOUN bracis-19045 220 31 for for ADP bracis-19045 220 32 dimension dimension NOUN bracis-19045 220 33 reduction reduction NOUN bracis-19045 220 34 . . PUNCT bracis-19045 221 1 arxiv arxiv PROPN bracis-19045 221 2 1802.03426 1802.03426 NUM bracis-19045 221 3 ( ( PUNCT bracis-19045 221 4 2020 2020 NUM bracis-19045 221 5 ) ) PUNCT bracis-19045 221 6 google google PROPN bracis-19045 221 7 scholar scholar NOUN bracis-19045 221 8     SPACE bracis-19045 221 9 de de X bracis-19045 221 10 ridder ridder PROPN bracis-19045 221 11 , , PUNCT bracis-19045 221 12 d. d. PROPN bracis-19045 221 13 , , PUNCT bracis-19045 221 14 duin duin PROPN bracis-19045 221 15 , , PUNCT bracis-19045 221 16 r.p r.p PROPN bracis-19045 221 17 . . PROPN bracis-19045 221 18 : : PUNCT bracis-19045 222 1 locally locally ADV bracis-19045 222 2 linear linear VERB bracis-19045 222 3 embedding embed VERB bracis-19045 222 4 for for ADP bracis-19045 222 5 classification classification NOUN bracis-19045 222 6 . . PUNCT bracis-19045 223 1 technical technical ADJ bracis-19045 223 2 report report PROPN bracis-19045 223 3 , , PUNCT bracis-19045 223 4 delft delft PROPN bracis-19045 223 5 university university PROPN bracis-19045 223 6 of of ADP bracis-19045 223 7 technology technology NOUN bracis-19045 223 8 ( ( PUNCT bracis-19045 223 9 2002 2002 NUM bracis-19045 223 10 ) ) PUNCT bracis-19045 223 11 google google PROPN bracis-19045 223 12 scholar scholar NOUN bracis-19045 223 13     SPACE bracis-19045 223 14 rousseeuw rousseeuw NOUN bracis-19045 223 15 , , PUNCT bracis-19045 223 16 p.j p.j PROPN bracis-19045 223 17 . . PROPN bracis-19045 223 18 : : PUNCT bracis-19045 223 19 silhouettes silhouette NOUN bracis-19045 223 20 : : PUNCT bracis-19045 223 21 a a DET bracis-19045 223 22 graphical graphical ADJ bracis-19045 223 23 aid aid NOUN bracis-19045 223 24 to to ADP bracis-19045 223 25 the the DET bracis-19045 223 26 interpretation interpretation NOUN bracis-19045 223 27 and and CCONJ bracis-19045 223 28 validation validation NOUN bracis-19045 223 29 of of ADP bracis-19045 223 30 cluster cluster NOUN bracis-19045 223 31 analysis analysis NOUN bracis-19045 223 32 . . PUNCT bracis-19045 224 1 j. j. PROPN bracis-19045 224 2 comput comput PROPN bracis-19045 224 3 . . PUNCT bracis-19045 225 1 appl appl PROPN bracis-19045 225 2 . . PROPN bracis-19045 225 3 math math NOUN bracis-19045 225 4 . . PUNCT bracis-19045 226 1 20 20 NUM bracis-19045 226 2 , , PUNCT bracis-19045 226 3 53–65 53–65 NUM bracis-19045 226 4 ( ( PUNCT bracis-19045 226 5 1987 1987 NUM bracis-19045 226 6 ) ) PUNCT bracis-19045 226 7 article article NOUN bracis-19045 226 8     SPACE bracis-19045 226 9 google google PROPN bracis-19045 226 10 scholar scholar NOUN bracis-19045 226 11     SPACE bracis-19045 226 12 roweis roweis NOUN bracis-19045 226 13 , , PUNCT bracis-19045 226 14 s. s. PROPN bracis-19045 226 15 , , PUNCT bracis-19045 226 16 saul saul PROPN bracis-19045 226 17 , , PUNCT bracis-19045 226 18 l. l. PROPN bracis-19045 226 19 : : PUNCT bracis-19045 226 20 nonlinear nonlinear ADJ bracis-19045 226 21 dimensionality dimensionality NOUN bracis-19045 226 22 reduction reduction NOUN bracis-19045 226 23 by by ADP bracis-19045 226 24 locally locally ADV bracis-19045 226 25 linear linear PROPN bracis-19045 226 26 embedding embed VERB bracis-19045 226 27 . . PUNCT bracis-19045 227 1 science science NOUN bracis-19045 227 2 290 290 NUM bracis-19045 227 3 , , PUNCT bracis-19045 227 4 2323–2326 2323–2326 NUM bracis-19045 227 5 ( ( PUNCT bracis-19045 227 6 2000 2000 NUM bracis-19045 227 7 ) ) PUNCT bracis-19045 227 8 article article NOUN bracis-19045 227 9     SPACE bracis-19045 227 10 google google PROPN bracis-19045 227 11 scholar scholar NOUN bracis-19045 227 12     SPACE bracis-19045 227 13 saul saul PROPN bracis-19045 227 14 , , PUNCT bracis-19045 227 15 l. l. PROPN bracis-19045 227 16 , , PUNCT bracis-19045 227 17 roweis roweis PROPN bracis-19045 227 18 , , PUNCT bracis-19045 227 19 s. s. PROPN bracis-19045 227 20 : : PUNCT bracis-19045 227 21 think think VERB bracis-19045 227 22 globally globally ADV bracis-19045 227 23 , , PUNCT bracis-19045 227 24 fit fit ADJ bracis-19045 227 25 locally locally ADV bracis-19045 227 26 : : PUNCT bracis-19045 227 27 unsupervised unsupervised ADJ bracis-19045 227 28 learning learning NOUN bracis-19045 227 29 of of ADP bracis-19045 227 30 low low ADJ bracis-19045 227 31 dimensional dimensional ADJ bracis-19045 227 32 manifolds manifold NOUN bracis-19045 227 33 . . PUNCT bracis-19045 228 1 j. j. PROPN bracis-19045 228 2 mach mach PROPN bracis-19045 228 3 . . PUNCT bracis-19045 229 1 learn learn VERB bracis-19045 229 2 . . PUNCT bracis-19045 230 1 res re NOUN bracis-19045 230 2 . . PROPN bracis-19045 231 1 4 4 NUM bracis-19045 231 2 , , PUNCT bracis-19045 231 3 119–155 119–155 NUM bracis-19045 231 4 ( ( PUNCT bracis-19045 231 5 2003 2003 NUM bracis-19045 231 6 ) ) PUNCT bracis-19045 231 7 mathscinet mathscinet NOUN bracis-19045 231 8     SPACE bracis-19045 231 9 math math NOUN bracis-19045 231 10     SPACE bracis-19045 231 11 google google PROPN bracis-19045 231 12 scholar scholar NOUN bracis-19045 231 13     SPACE bracis-19045 231 14 talmon talmon NOUN bracis-19045 231 15 , , PUNCT bracis-19045 231 16 r. r. PROPN bracis-19045 231 17 , , PUNCT bracis-19045 231 18 cohen cohen PROPN bracis-19045 231 19 , , PUNCT bracis-19045 231 20 i. i. PROPN bracis-19045 231 21 , , PUNCT bracis-19045 231 22 gannot gannot PROPN bracis-19045 231 23 , , PUNCT bracis-19045 231 24 s. s. PROPN bracis-19045 231 25 , , PUNCT bracis-19045 231 26 coifman coifman PROPN bracis-19045 231 27 , , PUNCT bracis-19045 231 28 r.r r.r PROPN bracis-19045 231 29 . . PROPN bracis-19045 231 30 : : PUNCT bracis-19045 232 1 diffusion diffusion NOUN bracis-19045 232 2 maps map NOUN bracis-19045 232 3 for for ADP bracis-19045 232 4 signal signal ADJ bracis-19045 232 5 processing processing NOUN bracis-19045 232 6 : : PUNCT bracis-19045 232 7 a a DET bracis-19045 232 8 deeper deep ADJ bracis-19045 232 9 look look NOUN bracis-19045 232 10 at at ADP bracis-19045 232 11 manifold manifold ADJ bracis-19045 232 12 - - PUNCT bracis-19045 232 13 learning learn VERB bracis-19045 232 14 techniques technique NOUN bracis-19045 232 15 based base VERB bracis-19045 232 16 on on ADP bracis-19045 232 17 kernels kernel NOUN bracis-19045 232 18 and and CCONJ bracis-19045 232 19 graphs graph NOUN bracis-19045 232 20 . . PUNCT bracis-19045 233 1 ieee ieee NOUN bracis-19045 233 2 signal signal NOUN bracis-19045 233 3 process process NOUN bracis-19045 233 4 . . PUNCT bracis-19045 234 1 mag mag INTJ bracis-19045 234 2 . . PUNCT bracis-19045 235 1 30(4 30(4 NUM bracis-19045 235 2 ) ) PUNCT bracis-19045 235 3 , , PUNCT bracis-19045 236 1 75–86 75–86 NUM bracis-19045 236 2 ( ( PUNCT bracis-19045 236 3 2013 2013 NUM bracis-19045 236 4 ) ) PUNCT bracis-19045 236 5 article article NOUN bracis-19045 236 6     SPACE bracis-19045 236 7 google google PROPN bracis-19045 236 8 scholar scholar NOUN bracis-19045 236 9     SPACE bracis-19045 236 10 tenenbaum tenenbaum PROPN bracis-19045 236 11 , , PUNCT bracis-19045 236 12 j.b j.b PROPN bracis-19045 236 13 . . PROPN bracis-19045 236 14 , , PUNCT bracis-19045 236 15 de de PROPN bracis-19045 236 16 silva silva PROPN bracis-19045 236 17 , , PUNCT bracis-19045 236 18 v. v. PROPN bracis-19045 236 19 , , PUNCT bracis-19045 236 20 langford langford PROPN bracis-19045 236 21 , , PUNCT bracis-19045 236 22 j.c j.c PROPN bracis-19045 236 23 . . PROPN bracis-19045 236 24 : : PUNCT bracis-19045 237 1 a a DET bracis-19045 237 2 global global ADJ bracis-19045 237 3 geometric geometric ADJ bracis-19045 237 4 framework framework NOUN bracis-19045 237 5 for for ADP bracis-19045 237 6 nonlinear nonlinear ADJ bracis-19045 237 7 dimensionality dimensionality NOUN bracis-19045 237 8 reduction reduction NOUN bracis-19045 237 9 . . PUNCT bracis-19045 238 1 science science NOUN bracis-19045 238 2 290 290 NUM bracis-19045 238 3 , , PUNCT bracis-19045 238 4 2319–2323 2319–2323 NUM bracis-19045 238 5 ( ( PUNCT bracis-19045 238 6 2000 2000 NUM bracis-19045 238 7 ) ) PUNCT bracis-19045 238 8 article article NOUN bracis-19045 238 9     SPACE bracis-19045 238 10 google google PROPN bracis-19045 238 11 scholar scholar NOUN bracis-19045 238 12     SPACE bracis-19045 238 13 wang wang PROPN bracis-19045 238 14 , , PUNCT bracis-19045 238 15 j. j. PROPN bracis-19045 238 16 : : PUNCT bracis-19045 238 17 hessian hessian NOUN bracis-19045 238 18 locally locally ADV bracis-19045 238 19 linear linear PROPN bracis-19045 238 20 embedding embed VERB bracis-19045 238 21 . . PUNCT bracis-19045 239 1 in in ADP bracis-19045 239 2 : : PUNCT bracis-19045 239 3 wang wang PROPN bracis-19045 239 4 , , PUNCT bracis-19045 239 5 j. j. PROPN bracis-19045 239 6 ( ( PUNCT bracis-19045 239 7 ed ed NOUN bracis-19045 239 8 . . PUNCT bracis-19045 239 9 ) ) PUNCT bracis-19045 239 10 geometric geometric ADJ bracis-19045 239 11 structure structure NOUN bracis-19045 239 12 of of ADP bracis-19045 239 13 high high ADJ bracis-19045 239 14 - - PUNCT bracis-19045 239 15 dimensional dimensional ADJ bracis-19045 239 16 data datum NOUN bracis-19045 239 17 and and CCONJ bracis-19045 239 18 dimensionality dimensionality NOUN bracis-19045 239 19 reduction reduction NOUN bracis-19045 239 20 , , PUNCT bracis-19045 239 21 pp pp ADJ bracis-19045 239 22 . . PUNCT bracis-19045 240 1 249–265 249–265 NUM bracis-19045 240 2 . . PUNCT bracis-19045 240 3 springer springer NOUN bracis-19045 240 4 , , PUNCT bracis-19045 240 5 heidelberg heidelberg PROPN bracis-19045 240 6 ( ( PUNCT bracis-19045 240 7 2012 2012 NUM bracis-19045 240 8 ) ) PUNCT bracis-19045 240 9 . . PUNCT bracis-19045 241 1 https://doi.org/10.1007/978-3-642-27497-8_13 https://doi.org/10.1007/978-3-642-27497-8_13 PROPN bracis-19045 241 2 wang wang PROPN bracis-19045 241 3 , , PUNCT bracis-19045 241 4 j. j. PROPN bracis-19045 241 5 , , PUNCT bracis-19045 241 6 wong wong PROPN bracis-19045 241 7 , , PUNCT bracis-19045 241 8 r.k r.k PROPN bracis-19045 241 9 . . PROPN bracis-19045 241 10 , , PUNCT bracis-19045 241 11 lee lee PROPN bracis-19045 241 12 , , PUNCT bracis-19045 241 13 t.c t.c PROPN bracis-19045 241 14 . . PROPN bracis-19045 241 15 : : PUNCT bracis-19045 242 1 locally locally ADV bracis-19045 242 2 linear linear VERB bracis-19045 242 3 embedding embed VERB bracis-19045 242 4 with with ADP bracis-19045 242 5 additive additive ADJ bracis-19045 242 6 noise noise NOUN bracis-19045 242 7 . . PUNCT bracis-19045 243 1 pattern pattern NOUN bracis-19045 243 2 recognit recognit VERB bracis-19045 243 3 . . PUNCT bracis-19045 244 1 lett lett PROPN bracis-19045 244 2 . . PUNCT bracis-19045 245 1 123 123 NUM bracis-19045 245 2 , , PUNCT bracis-19045 245 3 47–52 47–52 NUM bracis-19045 245 4 ( ( PUNCT bracis-19045 245 5 2019 2019 NUM bracis-19045 245 6 ) ) PUNCT bracis-19045 245 7 article article NOUN bracis-19045 245 8     SPACE bracis-19045 245 9 google google PROPN bracis-19045 245 10 scholar scholar NOUN bracis-19045 245 11     SPACE bracis-19045 245 12 xing xing PROPN bracis-19045 245 13 , , PUNCT bracis-19045 245 14 x. x. PROPN bracis-19045 245 15 , , PUNCT bracis-19045 245 16 du du PROPN bracis-19045 245 17 , , PUNCT bracis-19045 245 18 s. s. PROPN bracis-19045 245 19 , , PUNCT bracis-19045 245 20 wang wang PROPN bracis-19045 245 21 , , PUNCT bracis-19045 245 22 k. k. PROPN bracis-19045 245 23 : : PUNCT bracis-19045 245 24 robust robust ADJ bracis-19045 245 25 hessian hessian NOUN bracis-19045 245 26 locally locally ADV bracis-19045 245 27 linear linear VERB bracis-19045 245 28 embedding embed VERB bracis-19045 245 29 techniques technique NOUN bracis-19045 245 30 for for ADP bracis-19045 245 31 high high ADJ bracis-19045 245 32 - - PUNCT bracis-19045 245 33 dimensional dimensional ADJ bracis-19045 245 34 data datum NOUN bracis-19045 245 35 . . PUNCT bracis-19045 246 1 algorithms algorithm NOUN bracis-19045 246 2 9 9 NUM bracis-19045 246 3 , , PUNCT bracis-19045 246 4 36 36 NUM bracis-19045 246 5 ( ( PUNCT bracis-19045 246 6 2016 2016 NUM bracis-19045 246 7 ) ) PUNCT bracis-19045 246 8 article article NOUN bracis-19045 246 9     SPACE bracis-19045 246 10 mathscinet mathscinet NOUN bracis-19045 246 11     SPACE bracis-19045 246 12 google google PROPN bracis-19045 246 13 scholar scholar NOUN bracis-19045 246 14     SPACE bracis-19045 246 15 zhang zhang PROPN bracis-19045 246 16 , , PUNCT bracis-19045 246 17 z.y z.y PROPN bracis-19045 246 18 . . PROPN bracis-19045 246 19 , , PUNCT bracis-19045 246 20 zha zha PROPN bracis-19045 246 21 , , PUNCT bracis-19045 246 22 h.y h.y PROPN bracis-19045 246 23 . . PROPN bracis-19045 246 24 : : PUNCT bracis-19045 246 25 principal principal NOUN bracis-19045 246 26 manifolds manifold NOUN bracis-19045 246 27 and and CCONJ bracis-19045 246 28 nonlinear nonlinear ADJ bracis-19045 246 29 dimensionality dimensionality NOUN bracis-19045 246 30 reduction reduction NOUN bracis-19045 246 31 via via ADP bracis-19045 246 32 tangent tangent ADJ bracis-19045 246 33 space space NOUN bracis-19045 246 34 aligment aligment NOUN bracis-19045 246 35 . . PUNCT bracis-19045 247 1 siam siam PROPN bracis-19045 247 2 j. j. PROPN bracis-19045 247 3 sci sci PROPN bracis-19045 247 4 . . PUNCT bracis-19045 247 5 comput comput PROPN bracis-19045 247 6 . . PUNCT bracis-19045 248 1 26(1 26(1 NUM bracis-19045 248 2 ) ) PUNCT bracis-19045 248 3 , , PUNCT bracis-19045 248 4 313–338 313–338 NUM bracis-19045 248 5 ( ( PUNCT bracis-19045 248 6 2004 2004 NUM bracis-19045 248 7 ) ) PUNCT bracis-19045 248 8 article article NOUN bracis-19045 248 9     SPACE bracis-19045 248 10 mathscinet mathscinet NOUN bracis-19045 248 11     SPACE bracis-19045 248 12 google google PROPN bracis-19045 248 13 scholar scholar NOUN bracis-19045 248 14     SPACE bracis-19045 248 15 zhang zhang PROPN bracis-19045 248 16 , , PUNCT bracis-19045 248 17 z. z. PROPN bracis-19045 248 18 , , PUNCT bracis-19045 248 19 wang wang PROPN bracis-19045 248 20 , , PUNCT bracis-19045 248 21 j. j. PROPN bracis-19045 248 22 : : PUNCT bracis-19045 248 23 mlle mlle NOUN bracis-19045 248 24 : : PUNCT bracis-19045 248 25 modified modify VERB bracis-19045 248 26 locally locally ADV bracis-19045 248 27 linear linear ADJ bracis-19045 248 28 embedding embed VERB bracis-19045 248 29 using use VERB bracis-19045 248 30 multiple multiple ADJ bracis-19045 248 31 weights weight NOUN bracis-19045 248 32 . . PUNCT bracis-19045 249 1 in in ADP bracis-19045 249 2 : : PUNCT bracis-19045 249 3 schölkopf schölkopf PROPN bracis-19045 249 4 , , PUNCT bracis-19045 249 5 b. b. PROPN bracis-19045 249 6 , , PUNCT bracis-19045 249 7 platt platt PROPN bracis-19045 249 8 , , PUNCT bracis-19045 249 9 j.c j.c PROPN bracis-19045 249 10 . . PROPN bracis-19045 249 11 , , PUNCT bracis-19045 249 12 hofmann hofmann PROPN bracis-19045 249 13 , , PUNCT bracis-19045 249 14 t. t. PROPN bracis-19045 249 15 ( ( PUNCT bracis-19045 249 16 eds eds PROPN bracis-19045 249 17 . . PUNCT bracis-19045 249 18 ) ) PUNCT bracis-19045 249 19 advances advance NOUN bracis-19045 249 20 in in ADP bracis-19045 249 21 neural neural ADJ bracis-19045 249 22 information information NOUN bracis-19045 249 23 processing processing NOUN bracis-19045 249 24 systems system NOUN bracis-19045 249 25 19 19 NUM bracis-19045 249 26 , , PUNCT bracis-19045 249 27 proceedings proceeding NOUN bracis-19045 249 28 of of ADP bracis-19045 249 29 the the DET bracis-19045 249 30 20th 20th ADJ bracis-19045 249 31 conference conference NOUN bracis-19045 249 32 on on ADP bracis-19045 249 33 neural neural ADJ bracis-19045 249 34 information information NOUN bracis-19045 249 35 processing processing NOUN bracis-19045 249 36 systems system NOUN bracis-19045 249 37 , , PUNCT bracis-19045 249 38 vancouver vancouver PROPN bracis-19045 249 39 , , PUNCT bracis-19045 249 40 british british PROPN bracis-19045 249 41 columbia columbia PROPN bracis-19045 249 42 , , PUNCT bracis-19045 249 43 pp pp PROPN bracis-19045 249 44 . . PUNCT bracis-19045 250 1 1593–1600 1593–1600 NUM bracis-19045 250 2 ( ( PUNCT bracis-19045 250 3 2006 2006 NUM bracis-19045 250 4 ) ) PUNCT bracis-19045 250 5 google google PROPN bracis-19045 250 6 scholar scholar NOUN bracis-19045 250 7     SPACE bracis-19045 250 8 download download NOUN bracis-19045 250 9 references reference NOUN bracis-19045 250 10 acknowledgments acknowledgment NOUN bracis-19045 250 11 this this DET bracis-19045 250 12 study study NOUN bracis-19045 250 13 was be AUX bracis-19045 250 14 partially partially ADV bracis-19045 250 15 financed finance VERB bracis-19045 250 16 the the DET bracis-19045 250 17 coordenação coordenação NOUN bracis-19045 250 18 de de PROPN bracis-19045 250 19 aperfeiçoamento aperfeiçoamento PROPN bracis-19045 250 20 de de X bracis-19045 250 21 pessoal pessoal PROPN bracis-19045 250 22 de de PROPN bracis-19045 250 23 nível nível PROPN bracis-19045 250 24 superior superior PROPN bracis-19045 250 25 brasil brasil PROPN bracis-19045 250 26 ( ( PUNCT bracis-19045 250 27 capes cape NOUN bracis-19045 250 28 ) ) PUNCT bracis-19045 250 29 finance finance NOUN bracis-19045 250 30 code code NOUN bracis-19045 250 31 001 001 NUM bracis-19045 250 32 . . PUNCT bracis-19045 251 1 author author NOUN bracis-19045 251 2 information information NOUN bracis-19045 251 3 authors author NOUN bracis-19045 251 4 and and CCONJ bracis-19045 251 5 affiliations affiliation NOUN bracis-19045 251 6 computing computing PROPN bracis-19045 251 7 department department PROPN bracis-19045 251 8 , , PUNCT bracis-19045 251 9 federal federal ADJ bracis-19045 251 10 university university PROPN bracis-19045 251 11 of of ADP bracis-19045 251 12 são são PROPN bracis-19045 251 13 carlos carlos PROPN bracis-19045 251 14 , , PUNCT bracis-19045 251 15 são são PROPN bracis-19045 251 16 carlos carlos PROPN bracis-19045 251 17 , , PUNCT bracis-19045 251 18 sp sp PROPN bracis-19045 251 19 , , PUNCT bracis-19045 251 20 brazil brazil PROPN bracis-19045 251 21 alexandre alexandre PROPN bracis-19045 251 22 l. l. PROPN bracis-19045 251 23 m. m. PROPN bracis-19045 251 24 levada levada PROPN bracis-19045 251 25 department department PROPN bracis-19045 251 26 of of ADP bracis-19045 251 27 land land NOUN bracis-19045 251 28 economy economy NOUN bracis-19045 251 29 , , PUNCT bracis-19045 251 30 university university PROPN bracis-19045 251 31 of of ADP bracis-19045 251 32 cambridge cambridge PROPN bracis-19045 251 33 , , PUNCT bracis-19045 251 34 cambridge cambridge PROPN bracis-19045 251 35 , , PUNCT bracis-19045 251 36 uk uk PROPN bracis-19045 251 37 michel michel PROPN bracis-19045 251 38 f. f. PROPN bracis-19045 251 39 c. c. PROPN bracis-19045 251 40 haddad haddad PROPN bracis-19045 251 41 school school PROPN bracis-19045 251 42 of of ADP bracis-19045 251 43 business business NOUN bracis-19045 251 44 and and CCONJ bracis-19045 251 45 management management NOUN bracis-19045 251 46 , , PUNCT bracis-19045 251 47 queen queen PROPN bracis-19045 251 48 mary mary PROPN bracis-19045 251 49 university university PROPN bracis-19045 251 50 of of ADP bracis-19045 251 51 london london PROPN bracis-19045 251 52 , , PUNCT bracis-19045 251 53 london london PROPN bracis-19045 251 54 , , PUNCT bracis-19045 251 55 uk uk PROPN bracis-19045 251 56 michel michel PROPN bracis-19045 251 57 f. f. PROPN bracis-19045 251 58 c. c. PROPN bracis-19045 251 59 haddad haddad PROPN bracis-19045 251 60 authors authors PROPN bracis-19045 251 61 alexandre alexandre PROPN bracis-19045 251 62 l. l. PROPN bracis-19045 251 63 m. m. PROPN bracis-19045 251 64 levadaview levadaview PROPN bracis-19045 251 65 author author NOUN bracis-19045 251 66 publications publication NOUN bracis-19045 251 67 search search NOUN bracis-19045 251 68 author author NOUN bracis-19045 251 69 on on ADP bracis-19045 251 70 : : PUNCT bracis-19045 251 71 pubmed pubmed PROPN bracis-19045 251 72     SPACE bracis-19045 251 73 google google PROPN bracis-19045 251 74 scholar scholar PROPN bracis-19045 251 75 michel michel PROPN bracis-19045 251 76 f. f. PROPN bracis-19045 251 77 c. c. PROPN bracis-19045 251 78 haddadview haddadview PROPN bracis-19045 251 79 author author NOUN bracis-19045 251 80 publications publication VERB bracis-19045 251 81 search search NOUN bracis-19045 251 82 author author NOUN bracis-19045 251 83 on on ADP bracis-19045 251 84 : : PUNCT bracis-19045 251 85 pubmed pubmed PROPN bracis-19045 251 86     SPACE bracis-19045 251 87 google google PROPN bracis-19045 251 88 scholar scholar NOUN bracis-19045 251 89 editor editor NOUN bracis-19045 251 90 information information NOUN bracis-19045 251 91 editors editor NOUN bracis-19045 251 92 and and CCONJ bracis-19045 251 93 affiliations affiliation NOUN bracis-19045 251 94 universidade universidade PROPN bracis-19045 251 95 federal federal PROPN bracis-19045 251 96 de de X bracis-19045 251 97 sergipe sergipe PROPN bracis-19045 251 98 , , PUNCT bracis-19045 251 99 são são NOUN bracis-19045 251 100 cristóvão cristóvão PROPN bracis-19045 251 101 , , PUNCT bracis-19045 251 102 brazil brazil PROPN bracis-19045 251 103 andré andré PROPN bracis-19045 251 104 britto britto PROPN bracis-19045 251 105 universidade universidade PROPN bracis-19045 251 106 de de PROPN bracis-19045 251 107 são são PROPN bracis-19045 251 108 paulo paulo PROPN bracis-19045 251 109 , , PUNCT bracis-19045 251 110 são são PROPN bracis-19045 251 111 paulo paulo PROPN bracis-19045 251 112 , , PUNCT bracis-19045 251 113 brazil brazil PROPN bracis-19045 251 114 karina karina PROPN bracis-19045 251 115 valdivia valdivia PROPN bracis-19045 251 116 delgado delgado VERB bracis-19045 251 117 rights right NOUN bracis-19045 251 118 and and CCONJ bracis-19045 251 119 permissions permission NOUN bracis-19045 251 120 reprints reprint NOUN bracis-19045 251 121 and and CCONJ bracis-19045 251 122 permissions permission VERB bracis-19045 251 123 copyright copyright NOUN bracis-19045 251 124 information information NOUN bracis-19045 251 125 © © PROPN bracis-19045 251 126 2021 2021 NUM bracis-19045 251 127 springer springer NOUN bracis-19045 251 128 nature nature NOUN bracis-19045 251 129 switzerland switzerland PROPN bracis-19045 251 130 ag ag PROPN bracis-19045 251 131 about about ADP bracis-19045 251 132 this this DET bracis-19045 251 133 paper paper NOUN bracis-19045 251 134 cite cite VERB bracis-19045 251 135 this this DET bracis-19045 251 136 paper paper NOUN bracis-19045 251 137 levada levada PROPN bracis-19045 251 138 , , PUNCT bracis-19045 251 139 a.l.m a.l.m PROPN bracis-19045 251 140 . . PROPN bracis-19045 251 141 , , PUNCT bracis-19045 251 142 haddad haddad PROPN bracis-19045 251 143 , , PUNCT bracis-19045 251 144 m.f.c m.f.c PROPN bracis-19045 251 145 . . PROPN bracis-19045 252 1 ( ( PUNCT bracis-19045 252 2 2021 2021 NUM bracis-19045 252 3 ) ) PUNCT bracis-19045 252 4 . . PUNCT bracis-19045 253 1 a a DET bracis-19045 253 2 kullback kullback NOUN bracis-19045 253 3 - - PUNCT bracis-19045 253 4 leibler leibler NOUN bracis-19045 253 5 divergence divergence NOUN bracis-19045 253 6 - - PUNCT bracis-19045 253 7 based base VERB bracis-19045 253 8 locally locally ADV bracis-19045 253 9 linear linear ADJ bracis-19045 253 10 embedding embed VERB bracis-19045 253 11 method method NOUN bracis-19045 253 12 : : PUNCT bracis-19045 253 13 a a DET bracis-19045 253 14 novel novel ADJ bracis-19045 253 15 parametric parametric ADJ bracis-19045 253 16 approach approach NOUN bracis-19045 253 17 for for ADP bracis-19045 253 18 cluster cluster NOUN bracis-19045 253 19 analysis analysis NOUN bracis-19045 253 20 . . PUNCT bracis-19045 254 1 in in ADP bracis-19045 254 2 : : PUNCT bracis-19045 254 3 britto britto PROPN bracis-19045 254 4 , , PUNCT bracis-19045 254 5 a. a. PROPN bracis-19045 254 6 , , PUNCT bracis-19045 254 7 valdivia valdivia PROPN bracis-19045 254 8 delgado delgado PROPN bracis-19045 254 9 , , PUNCT bracis-19045 254 10 k. k. PROPN bracis-19045 254 11 ( ( PUNCT bracis-19045 254 12 eds eds PROPN bracis-19045 254 13 ) ) PUNCT bracis-19045 254 14 intelligent intelligent ADJ bracis-19045 254 15 systems system NOUN bracis-19045 254 16 . . PUNCT bracis-19045 255 1 bracis bracis PROPN bracis-19045 255 2 2021 2021 NUM bracis-19045 255 3 . . PUNCT bracis-19045 256 1 lecture lecture NOUN bracis-19045 256 2 notes note NOUN bracis-19045 256 3 in in ADP bracis-19045 256 4 computer computer NOUN bracis-19045 256 5 science science NOUN bracis-19045 256 6 ( ( PUNCT bracis-19045 256 7 ) ) PUNCT bracis-19045 256 8 , , PUNCT bracis-19045 256 9 vol vol NOUN bracis-19045 256 10 13073 13073 NUM bracis-19045 256 11 . . PUNCT bracis-19045 257 1 springer springer NOUN bracis-19045 257 2 , , PUNCT bracis-19045 257 3 cham cham PROPN bracis-19045 257 4 . . PUNCT bracis-19045 258 1 https://doi.org/10.1007/978-3-030-91702-9_27 https://doi.org/10.1007/978-3-030-91702-9_27 PROPN bracis-19045 258 2 download download NOUN bracis-19045 258 3 citation citation NOUN bracis-19045 258 4 .ris .ris PUNCT bracis-19045 259 1 .enw .enw PROPN bracis-19045 259 2 .bib .bib PUNCT bracis-19045 260 1 doi doi PROPN bracis-19045 260 2 : : PUNCT bracis-19045 260 3 https://doi.org/10.1007/978-3-030-91702-9_27 https://doi.org/10.1007/978-3-030-91702-9_27 PROPN bracis-19045 260 4 published publish VERB bracis-19045 260 5 : : PUNCT bracis-19045 260 6 28 28 NUM bracis-19045 260 7 november november PROPN bracis-19045 260 8 2021 2021 NUM bracis-19045 260 9 publisher publisher NOUN bracis-19045 260 10 name name NOUN bracis-19045 260 11 : : PUNCT bracis-19045 260 12 springer springer NOUN bracis-19045 260 13 , , PUNCT bracis-19045 260 14 cham cham PROPN bracis-19045 260 15 print print PROPN bracis-19045 260 16 isbn isbn PROPN bracis-19045 260 17 : : PUNCT bracis-19045 260 18 978 978 NUM bracis-19045 260 19 - - SYM bracis-19045 260 20 3 3 NUM bracis-19045 260 21 - - PUNCT bracis-19045 260 22 030 030 NUM bracis-19045 260 23 - - PUNCT bracis-19045 260 24 91701 91701 NUM bracis-19045 260 25 - - SYM bracis-19045 260 26 2 2 NUM bracis-19045 260 27 online online ADJ bracis-19045 260 28 isbn isbn NOUN bracis-19045 260 29 : : PUNCT bracis-19045 260 30 978 978 NUM bracis-19045 260 31 - - SYM bracis-19045 260 32 3 3 NUM bracis-19045 260 33 - - PUNCT bracis-19045 260 34 030 030 NUM bracis-19045 260 35 - - PUNCT bracis-19045 260 36 91702 91702 NUM bracis-19045 260 37 - - SYM bracis-19045 260 38 9 9 NUM bracis-19045 260 39 ebook ebook NOUN bracis-19045 260 40 packages package NOUN bracis-19045 260 41 : : PUNCT bracis-19045 260 42 computer computer NOUN bracis-19045 260 43 sciencecomputer sciencecomputer NOUN bracis-19045 260 44 science science NOUN bracis-19045 260 45 ( ( PUNCT bracis-19045 260 46 r0 r0 NOUN bracis-19045 260 47 ) ) PUNCT bracis-19045 260 48 share share VERB bracis-19045 260 49 this this DET bracis-19045 260 50 paper paper NOUN bracis-19045 260 51 anyone anyone PRON bracis-19045 260 52 you you PRON bracis-19045 260 53 share share VERB bracis-19045 260 54 the the DET bracis-19045 260 55 following follow VERB bracis-19045 260 56 link link NOUN bracis-19045 260 57 with with ADP bracis-19045 260 58 will will AUX bracis-19045 260 59 be be AUX bracis-19045 260 60 able able ADJ bracis-19045 260 61 to to PART bracis-19045 260 62 read read VERB bracis-19045 260 63 this this DET bracis-19045 260 64 content content NOUN bracis-19045 260 65 : : PUNCT bracis-19045 260 66 get get VERB bracis-19045 260 67 shareable shareable ADJ bracis-19045 260 68 linksorry linksorry NOUN bracis-19045 260 69 , , PUNCT bracis-19045 260 70 a a DET bracis-19045 260 71 shareable shareable ADJ bracis-19045 260 72 link link NOUN bracis-19045 260 73 is be AUX bracis-19045 260 74 not not PART bracis-19045 260 75 currently currently ADV bracis-19045 260 76 available available ADJ bracis-19045 260 77 for for ADP bracis-19045 260 78 this this DET bracis-19045 260 79 article article NOUN bracis-19045 260 80 . . PUNCT bracis-19045 261 1 copy copy VERB bracis-19045 261 2 shareable shareable ADJ bracis-19045 261 3 link link NOUN bracis-19045 261 4 to to PART bracis-19045 261 5 clipboard clipboard NOUN bracis-19045 261 6 provided provide VERB bracis-19045 261 7 by by ADP bracis-19045 261 8 the the DET bracis-19045 261 9 springer springer NOUN bracis-19045 261 10 nature nature PROPN bracis-19045 261 11 sharedit sharedit PROPN bracis-19045 261 12 content content NOUN bracis-19045 261 13 - - PUNCT bracis-19045 261 14 sharing share VERB bracis-19045 261 15 initiative initiative NOUN bracis-19045 261 16 keywords keyword NOUN bracis-19045 261 17 locally locally ADV bracis-19045 261 18 linear linear VERB bracis-19045 261 19 embedding embed VERB bracis-19045 261 20 metric metric ADJ bracis-19045 261 21 learning learning NOUN bracis-19045 261 22 kl kl PROPN bracis-19045 261 23 divergence divergence NOUN bracis-19045 261 24 publish publish NOUN bracis-19045 261 25 with with ADP bracis-19045 261 26 us us PROPN bracis-19045 261 27 policies policy NOUN bracis-19045 261 28 and and CCONJ bracis-19045 261 29 ethics ethic NOUN bracis-19045 261 30 profiles profile NOUN bracis-19045 261 31 michel michel PROPN bracis-19045 261 32 f. f. 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