EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5707
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

1

Proposal for a Generalized Convolution to Mitigate2

Heat Generation in Convolutional Neural Networks3

Hwajoon Kim1, Eunyoung Lim1, Byeongjae Kang1, Sunyoung Yeun1,∗
4

1 Kyungdong University, Kyungdong Univ. Rd. 27, Yangju, Gyeonggi, S. Korea5

6

Abstract. In convolutional neural networks (CNN), the problem of heat generation is becoming
a significant issue. This challenge can be mitigated through both hardware and software methods.
This study focuses on drastically reducing the amount of computations and, consequently, the heat
generation. Specifically, the proposed approach reduces computations by approximately m × 2n,
where m is the number of layers and n is the size of the node.

2020 Mathematics Subject Classifications: 68T01, 68T27, 44A357

Key Words and Phrases: Heat generation, convolutional neural networks, geralized convolution8

9

1. Introduction10

CNN can be understood as a specialized artificial intelligence framework particularly11

suited for processing image data. In CNN, convolution serves to transform the original12

image into a feature map using a matrix, while pooling acts as a form of data compres-13

sion. However, thses tasks often involve unnecessary computational processes, inspiring14

this research to explore the possibility of eliminating such inefficiencies. Consequently,15

this study proposes an algorithm that integrates convolution and pooling, traditionally16

performed as separate tasks, into a single, simplified operation. This integration aims to17

mitigate the heating problem caused by extensive computations in deep learning systems.18

Issues related to heating problems are discussed in [4].19

When CNN receives input, it generates multiple feature maps using the kernel matrix20

and then proceeds with classification. The kernel functions as the equivalent of weights21

in a neural network. Convolution, represented by a kernel matrix, operates as follows: by22

sweeping an n × n kernel matrix across the original image, the data is transforms into a23

new form. This process involves multiplying each n × n segment of the input matrix by24

the kernel matrix and summing all the resulting components, yielding the convolution.25

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5707

Email addresses: cellmath@gmail.com (Hj. Kim), minkimama0229@gmail.com (Ey. Lim),
kbj1216@kduniv.ac.kr (Bj. Kang), ysy3324@kduniv.ac.kr (Sy. Yeun)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



H. Kim et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5707 2 of 6

Deep learning concepts related to this study are discussed in [3], [2], [7], [8], [1], [9],26

and [10], while topic concerning integral transforms are presented in [5] and [6]. In this27

study, we propose a generalized convolution through straightforward examples, with the28

mathematical details to be elaborated on in future work.29

2. Proposal for a generalized convolution to mitigate heat generation in30

convolutional neural networks31

Existing neural networks update weights using the process

wa
ij = wb

ij − αew,

where α is the learning rate, wb represents the weight before the update, wa denotes the32

weight after the update, and ew represents the partial derivative of error e with respect to33

w. In CNN, convolution can be interpreted as a specific form of weight.34

Definition 1. The convolution of matrix A and matrix B is denoted by array multiplica-35

tion A ◦B.36

The following lemma shows the relationship between convolution in mathematics and37

convolution in AI.38

Lemma 1. A ◦B = tr(ABT ), where T is the transpose and tr is the trace.39

Proof. The detail is can be found in [2].40

The second step of CNN is the pooling. This is the work of reducing size, which is to
transform the image of the convolution into a small image by extracting a representative
from each part. For example, in short, if we have max-pooling a matrix(

1 2
3 4

)
,

we obtain a matrix (
4
)

of 1 row and 1 column(just a simple task of drawing the largest number). This pooling is41

simple, but it makes a contribution to enhance performance. Because, it is the principle42

that the resolution is increased when the screen is reduced. [8] has shown that features43

are invariant in fine translation and reduce noise. The currently widely used pooling44

method is max-pooling, which is quite simple as the above but has a reasonable effect.45

Now, let us define a generalized convolution as follows. Then, this definition can provide46

the simplicity of integrating the dualized convolution and max-pooling into one. Without47

further mention, sp, sc, ss and si refer to pooling size, convolution size, stride size, and48

image size, respectively.49

A matrix B is called a submatrix of A if B ⊂ A and satisfies the conditions of a matrix.50

51



H. Kim et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5707 3 of 6

Definition 2. Let A be an image data and Let M be a submatrix of A that contains the52

maximum element of A in the center. A generalized convolution incorporating convolution53

and max-pooling can be denoted by the maximum element of C ◦ M , where C is a given54

convolution.55

Let A. C, mb, and ma be the given image, convolution, the maximum element be-56

fore convolution, and the maximum element after convolution, respectively. Consider the57

submatrix {M |M ⊂ A} containing the maximum element mb. Note that M ⊆ A and58

sM = sc. A generalized convolution incorporating convolution and max-pooling can be59

denoted by ma ∈ (C ◦ M). This means the maximal element of the matrix created as a60

result of array multiplication of the image containing mb and the convolution.61

The significance of this definition is that there is no need to stack the convolution layer62

and the pooling layer as a pair, which is the conventional method. Instead, convolutional63

deep neural networks can be constructed using only a subset of the convolutional layers.64

This subset corresponds to the multiplication of maximum elements. To find the maximum65

element, simply select the matrix containing the largest number within n × n matrix66

(where n is fixed). In summary, CNN can be configured using a single task, generalized67

convolution, departing from the traditional approach that treats convolution and pooling68

as separate tasks.69

Comparison of generalized CNN algorithm and existing CNN algorithm70

A. Existing CNN algorithm71

(1) Input72

(2) Convolution73

(3) Pooling74

(4) Repeat (2) and (3) until the desired output is obtained.75

(5) Output76

B. generalized CNN algorithm77

(1) Input78

(2) Generalized convolution79

(3) Repeat (2) until the desired output is obtained.80

(4) Output81

In this deep learning method, generalized convolution applies convolution only to spe-82

cific nodes and does not require pooling work. This is similar to CNN applying only to83

specific nodes of the given image.84

Example. (Generalized convolution that greatly reduces computations) When the
original image is 

1 1 1 0 0
0 1 1 1 0
2 0 1 1 1
0 0 1 1 0
0 1 1 0 0

 ,



H. Kim et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5707 4 of 6

the convolved feature map is 6 3 4
2 4 3
4 3 4


for stride 1, where the kernel is 1 0 1

0 1 0
1 0 1

 .

Then if we do the max-pooling operation here, we get 6. Note that the convolution
operation is used 9 times here. If ss = 2, the convolved feature map changes to(

6 4
4 4

)
.

Of course, we get 6 after max-pooling, where the calculation of convolution is used 4 times.85

86

Next, let us consider the generalized convolution. The matrix from which the largest87

component can be obtained is the first 3 by 3 matrix. As a result of convolution, we get88

6. This is the same as the above result. Here, note that the convolution operation was89

performed only once and the pooling operation was not performed. As seen above, the90

generalized convolution becomes more effective as the size of the kernel matrix increases.91

It is only an estimate from which matrix can be obtained the largest component. This92

estimate is not necessarily accurate. Even a slight error will only bring about a slight93

difference in resolution. After choosing a matrix with large components, we just need to94

find the convolution. When the number of channels is multiple, the final feature map can95

be obtained by array addition of feature maps obtained from each channel.96

Example. Consider the following image

1 1 1 0 0 1 1 1 0 0
0 1 1 1 0 1 1 1 0 0
2 0 1 1 1 1 1 1 0 0
0 0 1 1 0 0 0 1 1 1
0 1 1 0 0 1 1 1 0 0
0 0 1 1 1 1 1 1 0 0
0 1 1 1 0 1 4 1 0 0
2 0 1 1 1 1 1 1 0 0
0 0 1 1 3 1 1 1 0 0
0 1 1 0 0 1 1 1 0 0


,



H. Kim et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5707 5 of 6

and let us ss = 1. Applying the given kernel matrix
1 1 1 0 0
0 1 1 1 0
2 0 1 1 1
0 0 1 1 0
0 1 1 0 0

 ,

we get the feature map 

17 9 8 9 12 12
9 9 10 12 11 11
10 9 11 15 14 9
10 9 11 12 14 9
10 13 17 18 14 11
15 12 14 16 17 14

 .

Applying max-pooling here, we get 18 for sp = 6. If the sp = 5, 4, and 3, we get the
matrices of (

18 18
18 18

)
,17 15 15

18 18 18
18 18 18

 ,

and 
17 15 15 15
11 15 15 15
17 18 18 18
17 18 18 18

 ,

respectively. The convolution operation was used 6× 6 = 36 times.97

In order to apply the generalized convolution here, let us select the following matrix:
1 0 0 1 1
1 1 1 1 1
1 1 0 1 4
1 1 1 1 1
1 1 3 1 1

 .

Array multiplication of the kernel matrix with this matrix gives 17. This is slightly
different from the result of the traditional method, which is 18. However, as mentioned
earlier, this small difference in resolution can be safely ignored. If ss = 2, the convolved
feature map is 17 8 12

10 11 14
10 17 14

 .



H. Kim et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5707 6 of 6

The result of max pooling is 17. Here convolution is used 3 × 3 = 9 times. Similarly, if98

the ss = 3, 4, 5 or 6, we also get a result of 17.99

This approach reduces the computational load by m × 2n in deep learning, where m100

is the number of layers and n is the size of the node. This reduction helps mitigate heat101

generation.102

Acknowledgements103

The first author (Hj. Kim) acknowledges the support of Kyungdong University Re-104

search Fund, 2024.105

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