From Global to Local Correlation: Geometric Decomposition of Statistical Inference

Understanding feature-outcome associations in high-dimensional data remainschallenging when relationships vary across subpopulations, yet standardmethods assuming global associations miss context-dependent patterns, reducingstatistical power and interpretability. We develop a geometric decompositionframework offering two strategies for partitioning inference problems intoregional analyses on data-derived Riemannian graphs. Gradient flowdecomposition uses path-monotonicity-validated discrete Morse theory topartition samples into gradient flow cells where outcomes exhibit monotonicbehavior. Co-monotonicity decomposition utilizes vertex-level coefficientsthat provide context-dependent versions of the classical Pearson correlation:these coefficients measure edge-based directional concordance between outcomeand features, or between feature pairs, defining embeddings of samples intoassociation space. These embeddings induce Riemannian k-NN graphs on whichbiclustering identifies co-monotonicity cells (coherent regions) and featuremodules. This extends naturally to multi-modal integration across multiplefeature sets. Both strategies apply independently or jointly, with Bayesianposterior sampling providing credible intervals.