Learning with Covariance Matrices: Principal Component Analysis Meets Learning with Graphs

This feature tutorial establishes the theoretical foundation for covariance neural networks (VNNs)—graph neural networks designed specifically to operate on covariance matrices as relational graphs. The authors formalize a conceptual equivalence between VNNs and classical principal component analysis (PCA), while establishing rigorous stability bounds under finite-sample perturbations and analyzing cross-dataset transferability. Grounded in signal processing and applied to brain age gap estimation from neuroimaging data, the work provides a principled theoretical justification for replacing traditional PCA workflows with graph architectures when data dependencies are statistical.