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.

翻译：图神经网络重新定义了我们对网络数据的建模与预测方式，但在选择用于信号建模的正确底层图结构方面缺乏共识。协方差神经网络通过使用样本协方差矩阵作为图移位算子解决了这一问题。本文通过构建密度矩阵改进了VNN的性能：在随机变量空间中，我们将样本协方差矩阵视为系统的准哈密顿量。关键之处在于，使用该密度矩阵作为GSO能够从不同尺度提取数据成分，从而增强可区分性与性能。我们证明该方法允许对网络的稳定性-可区分性权衡进行显式控制，相比VNN具有更强的噪声鲁棒性，并在底层协方差矩阵具有信息量的实际应用中表现更优。特别地，我们展示模型在受试者无关的脑机接口EEG运动想象分类任务中达到强性能，在比EEGnet速度更快的同时表现更优。这表明协方差密度神经网络为跨个体BCI迁移这一公认难题提供了基础性解决方案。