SHGR: A Generalized Maximal Correlation Coefficient

Samuel Stocksieker, Denys Pommeret

Keywords: Maximal Correlation, HGR, Non Linear Correlation, Neural Network

TL;DR: We introduce GHGR, a robust neural estimator of the Hirschfeld-Gebelein-Rényi (HGR) maximal correlation, which efficiently detects nonlinear dependencies, satisfies Rényi's axioms, and outperforms state-of-the-art methods.

Abstract: Traditional correlation measures, such as Pearson’s and Spearman’s coefficients, are limited in their ability to capture complex relationships, particularly nonlinear and multivariate dependencies. The Hirschfeld–Gebelein–Rényi (HGR) maximal correlation offers a powerful alternative by seeking the highest Pearson correlation attainable through nonlinear transformations of two random variables. However, estimating the HGR remains challenging due to the complexity of optimizing arbitrary nonlinear functions. We introduce a new coefficient inspired by the HGR but grounded in the Spearman rank correlation, which we call the Spearman HGR (SHGR). We propose a neural network-based estimator tailored to (i) bivariate correlation matrix, (ii) multivariate correlations between a set of variables and another one, and (iii) full correlation between two sets of variables. The SHGR satisfies Rényi’s axioms, effectively detects nonlinear dependencies, and demonstrates robustness to noise, outliers, and spurious correlations (hallucinations). Additionally, it achieves competitive computational efficiency through tailored neural architectures. Comprehensive Numerical experiments and feature selection tasks confirm that SHGR outperforms existing state-of-the-art methods.

Primary Area: General machine learning (supervised, unsupervised, online, active, etc.)

Submission Number: 7262