Covariance Density Neural Networks

TL;DR

Covariance Density Neural Networks address the problem of unknown graph structure by using the sample covariance as a quasi-Hamiltonian through a density matrix $\rho(\mathbf{C})$ with inverse temperature $\beta$. CDNNs enable multi-scale filtering via a density-based graph shift operator and learnable $\beta$-parametrized filter banks, yielding a controlled stability-discriminability trade-off. The authors introduce a multi-scale Von Neumann entropy CVNE for covariance matrices as an information-theoretic regularizer and prove permutation equivariance along with stability guarantees. Empirically, CDNNs surpass variance-based VNNs in financial forecasting and show strong transferability in EEG motor imagery BCIs, while offering faster inference, suggesting practical impact for real-time neural decoding and finance.

Abstract

Graph neural networks have re-defined how we model and predict on network data but there lacks a consensus on choosing the correct underlying graph structure on which to model signals. CoVariance Neural Networks (VNN) address this issue by using the sample covariance matrix as a Graph Shift Operator (GSO). Here, we improve on the performance of VNNs by constructing a Density Matrix where we consider the sample Covariance matrix as a quasi-Hamiltonian of the system in the space of random variables. Crucially, using this density matrix as the GSO allows components of the data to be extracted at different scales, allowing enhanced discriminability and performance. We show that this approach allows explicit control of the stability-discriminability trade-off of the network, provides enhanced robustness to noise compared to VNNs, and outperforms them in useful real-life applications where the underlying covariance matrix is informative. In particular, we show that our model can achieve strong performance in subject-independent Brain Computer Interface EEG motor imagery classification, outperforming EEGnet while being faster. This shows how covariance density neural networks provide a basis for the notoriously difficult task of transferability of BCIs when evaluated on unseen individuals.

Figures (15)

• Figure 1: Composite Lipschitz Condition Empirical Validation
• Figure 2: Discrimination of two singular Gaussians using CVNE. We sampled $N{=}5\!\times\!10^4$ triples $(X,Y,Z)$, estimated covariances on sliding windows ($W{=}128$) and computed $S_{\mathrm{naive}}$, $S_\beta$ ($\beta$ = 2). Background: 2D LDA decision regions. Dashed: best 1D splits—Naive fails (AUC 0.5), von Neumann succeeds (AUC 1.0).
• Figure 3: Contour Plot showing distribution of Covariance and Covariance Density Matrix on the Scalp at different scales. The top row indicates the averaged training covariance matrix (Excluding the Unseen Individual) and the bottom row indicates the unseen Individuals test covariance matrix. We can see that the training covariance matrix is relatively similar to the Unseen Individual and that different scales unveil further potential similarities thus increasing classification performance
• Figure 4: Ablation over different configurations for each subject.
• Figure 5: Mean Cohen’s kappa for each ablation configuration.

Theorems & Definitions (27)

• Definition 1: Graph Convolutional Filter
• Definition 2: Graph Fourier Transform (GFT) elvin
• Definition 3: Covariance and Sample Covariance
• Definition 4: coVariance Fourier Transform (VFT) cov
• Definition 5: Covariance Density Operator and Filter
• Definition 6: Covariance Density Perceptron
• Definition 7: Multi-Scale Covariance Density Filter Bank/Layer
• Theorem 1: Composite Lipschitz Conditions for covariance density Filters
• Theorem 2: Permutation Equivariance of the Covariance Density Filter
• Lemma 1: Error Bound for Covariance Density Matrix