Derivation of Back-propagation for Graph Convolutional Networks using Matrix Calculus and its Application to Explainable Artificial Intelligence

• URL: http://arxiv.org/abs/2408.01408v1
• Date: Fri, 2 Aug 2024 17:33:52 GMT
• Title: Derivation of Back-propagation for Graph Convolutional Networks using Matrix Calculus and its Application to Explainable Artificial Intelligence
• Abstract summary: This paper provides a comprehensive and detailed derivation of the backpropagation algorithm for graph convolutional neural networks. The derivation is extended to include arbitrary element-wise activation functions and an arbitrary number of layers. We show how the derived closed-form solution can facilitate the development of explainable AI and sensitivity analysis.
• Score: 3.4644295862912866
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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