Neural Tangent Kernels Motivate Graph Neural Networks with Cross-Covariance Graphs
This work addresses the challenge of enhancing GNN performance for tasks like time series prediction, though it appears incremental by applying NTK theory to a specific domain.
The paper tackled the problem of improving graph neural networks (GNNs) by analyzing neural tangent kernels (NTKs) to optimize graph representations, showing that using cross-covariance as the graph shift operator outperforms input-only covariance in a multi-variate time series prediction task.
Neural tangent kernels (NTKs) provide a theoretical regime to analyze the learning and generalization behavior of over-parametrized neural networks. For a supervised learning task, the association between the eigenvectors of the NTK kernel and given data (a concept referred to as alignment in this paper) can govern the rate of convergence of gradient descent, as well as generalization to unseen data. Building upon this concept, we investigate NTKs and alignment in the context of graph neural networks (GNNs), where our analysis reveals that optimizing alignment translates to optimizing the graph representation or the graph shift operator in a GNN. Our results further establish the theoretical guarantees on the optimality of the alignment for a two-layer GNN and these guarantees are characterized by the graph shift operator being a function of the cross-covariance between the input and the output data. The theoretical insights drawn from the analysis of NTKs are validated by our experiments focused on a multi-variate time series prediction task for a publicly available dataset. Specifically, they demonstrate that GNNs with cross-covariance as the graph shift operator indeed outperform those that operate on the covariance matrix from only the input data.