[论文] Learning with Covariance Matrices: Principal Component Analysis Meets Learning with Graphs (arXiv:2609.10490)

论文概要

研究领域: ML
作者: Saurabh Sihag, Andrea Cavallo, Elvin Isufi, Gonzalo Mateos, Alejandro Ribeiro
发布时间: 2026-09-09
arXiv: 2609.10490

中文摘要

本文综述了协方差神经网络(VNNs)的理论基础，即运行在将协方差矩阵视为图结构的图神经网络。协方差矩阵在各个领域无处不在，GNN的部署经常利用成对统计依赖的图。本文通过数学分析带来多种新颖理论洞见：(i) VNN与基于主成分分析(PCA)的信息处理之间的概念等价性；(ii) 有限样本诱导协方差矩阵扰动存在时预测结果的精细稳定性界；(iii) VNN跨多尺度数据集的迁移性的精细刻画。这些理论洞见为在协方差矩阵是数据结构有用描述符的应用中采用VNN而非PCA提供了原理性依据。

原文摘要

This feature article provides an overview of the theoretical foundations for coVariance neural networks (VNNs), i.e., graph neural networks (GNNs) operating on covariance matrices as graphs. Covariance matrices are ubiquitous across domains, and hence, the deployment of GNNs often leverages graphs of pairwise statistical dependencies. Existing theoretical contributions on GNNs consider abstract graph representations and cannot accommodate the data-driven nuances associated with covariance matrices. This tutorial brings into focus various novel theoretical insights via mathematical analyses of VNNs that have broad signal processing implications, including: (i) a conceptual equivalence between VNNs and principal component analysis (PCA)-based information processing; (ii) refined stability bounds on predictive outcomes in the presence of finite sample-induced covariance matrix perturbations; and (iii) refined characterization of transferability of VNNs across multiscale datasets. The theoretical insights discussed herein provide the underlying principles and justification towards adopting VNNs over workhorse PCA-based learning pipelines, in applications where covariance matrices are useful descriptors of data structure. We also convey how impact of these foundational advances permeates to \textit{principled} designs and applications of learning methods across broad domains where covariance matrices emerge. Notably, we elucidate the conceptual insights facilitated by VNNs to the specific task of characterizing brain age gap for neurodegenerative conditions using neuroimaging datasets, a timely problem in computational neuroscience. Broader impacts to other application domains are discussed as well.

自动采集于 2026-09-11

#论文 #arXiv #ML #小凯