SHGR: A Generalized Maximal Correlation Coefficient
Samuel Stocksieker, Denys Pommeret
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 matrices, (ii) multivariate correlations between a set of predictors and a target, and (iii) full correlations between two sets of variables. The SHGR satisfies Rényi's axioms, effectively detects nonlinear dependencies, and demonstrates robustness to noise, outliers, and spurious correlations. Additionally, it achieves competitive computational efficiency through tailored neural architectures. Numerical experiments and feature selection tasks confirm that the SHGR outperforms existing state-of-the-art methods.