Bayesian Matrix-Valued Graphs for Context-Dependent Multivariate Relationships
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.
Related
- Matrix roots of order n of a multivariate polynomial
- Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices
- ChorusTIC: Training-Free Multivariate Time Series Classification via Chorus In-Context Learning
- Salient Bayesian Clustering for Proportional Data via a Multivariate Beta Mixture Model
- Bayesian model averaging of risk set probabilities using a geometric representation of multivariate extremes
BibTeX
@misc{dey2026bayesian,
title = {{Bayesian Matrix-Valued Graphs for Context-Dependent Multivariate Relationships}},
author = {Papri Dey},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08055},
eprint = {2609.08055},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08055},
}
RIS
TY - UNPB
TI - Bayesian Matrix-Valued Graphs for Context-Dependent Multivariate Relationships
AU - Papri Dey
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.08055
UR - https://arxiv.org/abs/2609.08055
AB - 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.
ER -