Geometric Decomposition of Statistical Inference through Gradient Flow and Co-Monotonicity Measures

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 basins where outcomes exhibit monotonic behavior.Co-monotonicity decomposition leverages association structure: vertex-levelcoefficients measuring directional concordance between outcome and features,or between feature pairs, define embeddings of samples into association space.These embeddings induce Riemannian k-NN graphs on which biclusteringidentifies co-monotonicity cells (coherent regions) and feature modules. Thisextends naturally to multi-modal integration across multiple feature sets.Both strategies apply independently or jointly, with Bayesian posteriorsampling providing credible intervals.