coVariance Neural Networks
This addresses instability issues in data analysis for domains using covariance matrices, offering a more robust alternative to PCA-based approaches.
The paper tackles the instability of PCA-based methods when dealing with close eigenvalues by proposing coVariance Neural Networks (VNNs), which operate on sample covariance matrices as graphs, and shows that VNNs achieve more stable performance in experiments on real-world datasets.
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.