coVariance Neural Networks

TL;DR

This paper introduces coVariance Neural Networks (VNNs), a novel GNN architecture that operates on covariance matrices, demonstrating enhanced stability and transferability over PCA-based methods through theoretical analysis and real-world experiments.

Contribution

The paper proposes VNNs that leverage covariance matrices in GNNs, providing theoretical stability guarantees and demonstrating improved robustness and transferability over PCA-based approaches.

Findings

VNNs are more stable than PCA-based methods under covariance perturbations.

VNNs exhibit transferability across datasets with different covariance matrix dimensions.

Experimental results validate the theoretical stability and transferability advantages of VNNs.

Abstract

Graph neural networks (GNN) are an effective framework that exploit inter-relationships within graph-structured data for learning. Principal component analysis (PCA) involves the projection of data on the eigenspace of the covariance matrix and draws similarities with the graph convolutional filters in GNNs. Motivated by this observation, we study a GNN architecture, called coVariance neural network (VNN), that operates on sample covariance matrices as graphs. We theoretically establish the stability of VNNs to perturbations in the covariance matrix, thus, implying an advantage over standard PCA-based data analysis approaches that are prone to instability due to principal components associated with close eigenvalues. Our experiments on real-world datasets validate our theoretical results and show that VNN performance is indeed more stable than PCA-based statistical approaches. Moreover, our experiments on multi-resolution datasets also demonstrate that VNNs are amenable to transferability of performance over covariance matrices of different dimensions; a feature that is infeasible for PCA-based approaches.