Derivation of Back-propagation for Graph Convolutional Networks using Matrix Calculus and its Application to Explainable Artificial Intelligence
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910552662474752 |
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| author | Hsiao, Yen-Che Yue, Rongting Dutta, Abhishek |
| author_facet | Hsiao, Yen-Che Yue, Rongting Dutta, Abhishek |
| contents | This paper provides a comprehensive and detailed derivation of the backpropagation algorithm for graph convolutional neural networks using matrix calculus. The derivation is extended to include arbitrary element-wise activation functions and an arbitrary number of layers. The study addresses two fundamental problems, namely node classification and link prediction. To validate our method, we compare it with reverse-mode automatic differentiation. The experimental results demonstrate that the median sum of squared errors of the updated weight matrices, when comparing our method to the approach using reverse-mode automatic differentiation, falls within the range of $10^{-18}$ to $10^{-14}$. These outcomes are obtained from conducting experiments on a five-layer graph convolutional network, applied to a node classification problem on Zachary's karate club social network and a link prediction problem on a drug-drug interaction network. Finally, we show how the derived closed-form solution can facilitate the development of explainable AI and sensitivity analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01408 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derivation of Back-propagation for Graph Convolutional Networks using Matrix Calculus and its Application to Explainable Artificial Intelligence Hsiao, Yen-Che Yue, Rongting Dutta, Abhishek Machine Learning This paper provides a comprehensive and detailed derivation of the backpropagation algorithm for graph convolutional neural networks using matrix calculus. The derivation is extended to include arbitrary element-wise activation functions and an arbitrary number of layers. The study addresses two fundamental problems, namely node classification and link prediction. To validate our method, we compare it with reverse-mode automatic differentiation. The experimental results demonstrate that the median sum of squared errors of the updated weight matrices, when comparing our method to the approach using reverse-mode automatic differentiation, falls within the range of $10^{-18}$ to $10^{-14}$. These outcomes are obtained from conducting experiments on a five-layer graph convolutional network, applied to a node classification problem on Zachary's karate club social network and a link prediction problem on a drug-drug interaction network. Finally, we show how the derived closed-form solution can facilitate the development of explainable AI and sensitivity analysis. |
| title | Derivation of Back-propagation for Graph Convolutional Networks using Matrix Calculus and its Application to Explainable Artificial Intelligence |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2408.01408 |