AI 中文总结
本文提出贝叶斯矩阵值图（BMVG）框架，用SPD矩阵建模边权重，通过黎曼几何量化上下文相关的多元关系变化，在天气和基因数据中验证了其有效性和可解释性。
AI 中文摘要
许多科学图中的每个节点都关联多个变量，因此单一的标量边权重无法描述方向依赖的交互。我们用对称正定（SPD）矩阵对每条边进行建模，并推断矩阵值图几何结构上的后验分布，我们称之为贝叶斯矩阵值图（BMVG）。我们探究这些交互如何在不同上下文之间重新配置：变化有多大，以及哪些多元方向增强或减弱。由仿射不变黎曼度量（AIRM）诱导的测地距离量化了变形幅度，广义特征值解析了其有符号的方向。与融合图形套索、贝叶斯多重高斯图模型和共同主成分相比，BMVG在全局精度恢复上具有竞争力，同时保留了可识别的矩阵值边结构，并准确恢复边级变形方向。在受控的已知真值实验中，它随着样本量的增加解析结构变化，包括那些不改变普通特征值的定向变化。在一年湾区天气数据中，12小时变化的几何结构对空间耦合的重新配置程度与整个季节的差异相当。在TCGA-BRCA中，雌激素受体（ER）相关的重新配置集中于特定的基因模块对，并在图支架稀疏化和去除亚组均值差异后持续存在。这些结果确立了后验矩阵值边几何作为量化和解释上下文相关多元重新配置的统一框架。
英文摘要
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.
