Bayesian Matrix-Valued Graphs for Context-Dependent Multivariate Relationships

2026-09-07

AI Summary

本文提出了一种贝叶斯矩阵值图模型，用于描述多变量关系在不同上下文中的变化。该方法通过使用对称正定矩阵表示边，并利用仿射不变黎曼度量来量化变形幅度和方向。

Abstract

Many scientific graphs attach several variables to each node, so a single scalar edge weight cannot describe direction-dependent interactions. We model each edge by a symmetric positive-definite (SPD) matrix and infer a posterior over matrix-valued graph geometries, which we call the Bayesian matrix-valued graph (BMVG). We ask how these interactions reconfigure across contexts: how large the change is and which multivariate directions strengthen or weaken. The geodesic distance induced by the affine-invariant Riemannian metric (AIRM) quantifies deformation magnitude and generalized eigenvalues resolve its signed directions.Against fused graphical lasso, Bayesian multiple-GGM, and common principal components, BMVG is competitive on global precision recovery while retaining identifiable matrix-valued edge structure and accurately recovering edge-level deformation directions. In controlled known-truth experiments, it resolves structural change with increasing sample size, including orientation changes that leave ordinary eigenvalues unchanged. In one year of Bay Area weather data, the geometry of 12-hour change reconfigures spatial coupling about as much as whole seasons differ. In TCGA-BRCA, estrogen-receptor (ER)-associated reconfiguration concentrates on specific gene-module pairs and persists under graph-scaffold sparsification and removal of subgroup mean differences. These results establish posterior matrix-valued edge geometry as a unified framework for quantifying and interpreting context-dependent multivariate reconfiguration.

Problem

• multivariate relationships
• context-dependent
• matrix-valued graph
• reconfiguration
• geodesic distance

Innovation

• Bayesian matrix-valued graph (BMVG)
• symmetric positive-definite (SPD) matrix
• affine-invariant Riemannian metric (AIRM)
• generalized eigenvalues
• context-dependent multivariate relationships