无序对空间上的协方差核：理论与网络值数据的应用
Covariance Kernels on Unordered Pair Spaces: Theory and Applications to Network-Valued Data
- St. Edward’s University（圣爱德华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种无序关系协方差框架，将对象几何转移到关系上，并应用于脑连接组数据，发现大连接组有效协方差维度较小，且高解释协方差不足以选择低秩表示。
AI中文摘要：
许多科学问题是关系型的：感兴趣的量是两个对象之间的联系，而关于对象本身相似性的信息是可获得的。我们开发了一个用于无序关系的协方差框架，该框架将对象级几何转移到它们形成的关系上，同时保持端点身份和对排序的不变性。基于对称的成对核表示，我们为无向网络中使用的无环域建立了理论。我们建立了在移除自配对后的谱交错和迹损失结果，将关系谱与正则化和风险联系起来，推导出谱截断的精确推断误差，并量化底层几何的扰动如何传播到配对协方差和估计。模拟展示了结构化借用何时能改善估计，以及几何错误设定如何削弱这种收益。我们将该框架应用于自闭症神经影像学，使用来自自闭症脑成像数据交换（ABIDE）的静息态功能连接数据。在116个区域的分割和6,670个独特连接下，该应用表明一个大的连接组可以具有更小的有效协方差维度。它还表明，当推断准确性是目标时，仅高解释协方差不足以选择低秩表示。该框架为统计单元是无序关系时的协方差和正则化提供了原则性基础。
英文摘要：
Many scientific problems are relational: the quantity of interest is a connection between two objects, while information about similarity is available for the objects themselves. We develop a covariance framework for unordered relationships that transfers object-level geometry to the relations they form while preserving endpoint identity and invariance to ordering. Building on symmetric pairwise-kernel representations, we develop theory for the loop-free domains used in undirected networks. We establish spectral interlacing and trace-loss results after self-pairs are removed, connect the relational spectrum to regularization and risk, derive an exact inferential error for spectral truncation, and quantify how perturbations of the underlying geometry propagate to pair covariance and estimation. Simulations show when structured borrowing improves estimation and how geometric misspecification can erode that benefit. We apply the framework to autism neuroimaging using resting-state functional-connectivity data from the Autism Brain Imaging Data Exchange (ABIDE). With a 116-region parcellation and 6,670 unique connections, the application shows that a large connectome can have a much smaller effective covariance dimension. It also demonstrates that high explained covariance alone is insufficient for choosing a low-rank representation when inferential accuracy is the goal. The framework provides a principled foundation for covariance and regularization when the statistical units are unordered relationships.
