发表机构
University at Albany, SUNY; Delft University of Technology; University of Rochester; University of Pennsylvania（纽约州立大学奥尔巴尼分校; 代尔夫特理工大学; 罗切斯特大学; 宾夕法尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文综述协方差神经网络的理论基础，阐明其与主成分分析的概念等价性、稳定性界限及跨尺度可迁移性，并展示其在脑年龄差距预测等任务中的优势。
AI 中文摘要
这篇专题文章概述了协方差神经网络（VNNs）的理论基础，即作用于协方差矩阵作为图的图神经网络（GNNs）。协方差矩阵在各个领域普遍存在，因此，GNNs的部署通常利用成对统计依赖的图。现有的关于GNNs的理论贡献考虑的是抽象的图表示，无法适应与协方差矩阵相关的数据驱动细微差别。本教程通过对VNNs的数学分析，聚焦多种新颖的理论见解，这些见解具有广泛的信号处理意义，包括：（i）VNNs与基于主成分分析（PCA）的信息处理之间的概念等价性；（ii）在有限样本引起的协方差矩阵扰动存在的情况下，预测结果的精细化稳定性界限；（iii）VNNs跨多尺度数据集可迁移性的精细化表征。本文讨论的理论见解为在协方差矩阵作为数据结构有用描述符的应用中，采用VNNs而非主流的基于PCA的学习流程提供了基本原则和理由。我们还传达了这些基础性进展的影响如何渗透到协方差矩阵出现的广泛领域中学习方法的原则性设计和应用。值得注意的是，我们阐明了VNNs为使用神经影像数据集表征神经退行性疾病脑年龄差距这一特定任务所提供的概念性见解，这是计算神经科学中一个及时的问题。此外，还讨论了对其他应用领域的更广泛影响。
英文摘要
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.
CommentsAccepted for publication in IEEE Signal Processing Magazine
