Choquet integral based on copulas and its application in neural networks

Hong Yang; Jiaxue Wei; Hong Yang; Jiaxue Wei

doi:10.3934/nhm.2026044

Networks and Heterogeneous Media

2026, Volume 21, Issue 3: 1069-1098. doi: 10.3934/nhm.2026044

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Choquet integral based on copulas and its application in neural networks

Hong Yang ^,,
Jiaxue Wei

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, PR China

Received: 25 December 2025 Revised: 13 April 2026 Accepted: 08 May 2026 Published: 21 May 2026

Usually, convolutional neural networks (CNNs) reduce the features extracted from convolutional layers by adopting the maximum value or arithmetic average method, which is called the pooling process. However, these two pooling methods overlook the spatial dependencies between image features. The Choquet integral is a nonlinear integral, which addresses this limitation by incorporating weighted aggregation via a fuzzy measure, enabling it to effectively model interactions between variables. This capability is essential for processing data with complex dependency structures. Therefore, this paper proposes a novel Choquet integral pooling method for feature extraction in the pooling layer. By conducting experiments on datasets, we demonstrated the effectiveness of the proposed method and compared it with existing techniques. The experimental results indicated that for cases where image features are less prominent and exhibit spatial dependencies, the Choquet integral pooling outperforms traditional pooling methods and demonstrates greater robustness. To ensure the rationality and validity of applying the Choquet integral in CNNs, we conducted a more in-depth study of the Choquet integral based on a copula, that is, the $ C_{C^{-}} $integral, including its averaging, idempotence, translation invariance, and positive homogeneity. This research not only enriches the application fields of the Choquet integral but also provides new theoretical support and technical paths for the research of neural networks.
- fuzzy analysis,
- Choquet integral,
- $ C_{C^{-}} $integral,
- convolutional neural networks
Citation: Hong Yang, Jiaxue Wei. Choquet integral based on copulas and its application in neural networks[J]. Networks and Heterogeneous Media, 2026, 21(3): 1069-1098. doi: 10.3934/nhm.2026044

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Abstract

Usually, convolutional neural networks (CNNs) reduce the features extracted from convolutional layers by adopting the maximum value or arithmetic average method, which is called the pooling process. However, these two pooling methods overlook the spatial dependencies between image features. The Choquet integral is a nonlinear integral, which addresses this limitation by incorporating weighted aggregation via a fuzzy measure, enabling it to effectively model interactions between variables. This capability is essential for processing data with complex dependency structures. Therefore, this paper proposes a novel Choquet integral pooling method for feature extraction in the pooling layer. By conducting experiments on datasets, we demonstrated the effectiveness of the proposed method and compared it with existing techniques. The experimental results indicated that for cases where image features are less prominent and exhibit spatial dependencies, the Choquet integral pooling outperforms traditional pooling methods and demonstrates greater robustness. To ensure the rationality and validity of applying the Choquet integral in CNNs, we conducted a more in-depth study of the Choquet integral based on a copula, that is, the $ C_{C^{-}} $integral, including its averaging, idempotence, translation invariance, and positive homogeneity. This research not only enriches the application fields of the Choquet integral but also provides new theoretical support and technical paths for the research of neural networks.

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