From Global to Local Correlation: Geometric Decomposition of Statistical Inference

TL;DR

This paper introduces a geometric decomposition framework for high-dimensional data analysis, enabling context-dependent regional inference through Riemannian graph partitioning and association space embeddings.

Contribution

It proposes two novel strategies—gradient flow and co-monotonicity decomposition—for regional analysis of feature-outcome associations in high-dimensional data.

Findings

Gradient flow decomposition partitions data into monotonic regions.

Co-monotonicity coefficients extend Pearson correlation to context-dependent measures.

Framework supports multi-modal data integration and Bayesian inference.

Abstract

Understanding feature-outcome associations in high-dimensional data remains challenging when relationships vary across subpopulations, yet standard methods assuming global associations miss context-dependent patterns, reducing statistical power and interpretability. We develop a geometric decomposition framework offering two strategies for partitioning inference problems into regional analyses on data-derived Riemannian graphs. Gradient flow decomposition uses path-monotonicity-validated discrete Morse theory to partition samples into gradient flow cells where outcomes exhibit monotonic behavior. Co-monotonicity decomposition utilizes vertex-level coefficients that provide context-dependent versions of the classical Pearson correlation: these coefficients measure edge-based directional concordance between outcome and features, or between feature pairs, defining embeddings of samples into association space. These embeddings induce Riemannian k-NN graphs on which biclustering identifies co-monotonicity cells (coherent regions) and feature modules. This extends naturally to multi-modal integration across multiple feature sets. Both strategies apply independently or jointly, with Bayesian posterior sampling providing credible intervals.