Derivation of Back-propagation for Graph Convolutional Networks using Matrix Calculus and its Application to Explainable Artificial Intelligence
<|reference_start|>Derivation of Back-propagation for Graph Convolutional Networks using Matrix Calculus and its Application to Explainable Artificial Intelligence: 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.<|reference_end|>
@article{hsiao2024derivation,
    title={Derivation of Back-propagation for Graph Convolutional Networks using
  Matrix Calculus and its Application to Explainable Artificial Intelligence},
    author={Yen-Che Hsiao, Rongting Yue, Abhishek Dutta},
    journal={arXiv preprint arXiv:2408.01408},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.01408},
    primaryClass={cs.LG}
}