Matrix-valued graphs reveal how multivariate relationships change across contexts
Bayesian Matrix-Valued Graphs for Context-Dependent Multivariate Relationships
Machine Learning
Summary
Many scientific studies use networks where each point has several measurements, which makes it hard to describe how connections between points change under different conditions. The authors developed a method that represents each connection as a matrix rather than a single number, capturing more detailed interactions. Their approach measures how these connections reshape and which aspects get stronger or weaker in different scenarios. They tested their method on simulated data, weather patterns, and gene data, showing it can detect complex changes that other methods might miss.
graph theorymatrix-valued edgesBayesian inferencesymmetric positive-definite matricesRiemannian geometrymultivariate relationshipsgraphical modelscontext-dependent changes
Authors
Papri Dey
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.