A Generalization of the Pearson Correlation to Riemannian Manifolds

TL;DR

This paper introduces a Riemannian manifold-based generalization of Pearson correlation, enabling correlation analysis in complex non-linear statistical models represented as Riemannian manifolds.

Contribution

It develops a novel Riemannian Pearson correlation measure that extends traditional correlation to non-linear, manifold-structured data, bridging geometry and statistical analysis.

Findings

Defines Riemannian Pearson correlation on tangent spaces.

Derives a nonlinear generalization for principal manifolds.

Provides a framework for correlation analysis on Riemannian manifolds.

Abstract

The increasing application of deep-learning is accompanied by a shift towards highly non-linear statistical models. In terms of their geometry it is natural to identify these models with Riemannian manifolds. The further analysis of the statistical models therefore raises the issue of a correlation measure, that in the cutting planes of the tangent spaces equals the respective Pearson correlation and extends to a correlation measure that is normalized with respect to the underlying manifold. In this purpose the article reconstitutes elementary properties of the Pearson correlation to successively derive a linear generalization to multiple dimensions and thereupon a nonlinear generalization to principal manifolds, given by the Riemann-Pearson Correlation.