Generalized infinite dimensional Alpha-Procrustes based geometries

TL;DR

This paper extends the Alpha-Procrustes family of Riemannian metrics to infinite dimensions by incorporating generalized Bures-Wasserstein, Log-Euclidean, and Wasserstein distances, enabling more robust comparisons of high-dimensional data.

Contribution

It introduces a formalism based on unitized Hilbert-Schmidt operators and an extended Mahalanobis norm to generalize key metrics in infinite-dimensional spaces, with a learnable regularization parameter.

Findings

Improved performance in dataset comparison benchmarks

Robustness in high-dimensional and scale-varying scenarios

Theoretical foundation for advanced geometric methods

Abstract

This work extends the recently introduced Alpha-Procrustes family of Riemannian metrics for symmetric positive definite (SPD) matrices by incorporating generalized versions of the Bures-Wasserstein (GBW), Log-Euclidean, and Wasserstein distances. While the Alpha-Procrustes framework has unified many classical metrics in both finite- and infinite- dimensional settings, it previously lacked the structural components necessary to realize these generalized forms. We introduce a formalism based on unitized Hilbert-Schmidt operators and an extended Mahalanobis norm that allows the construction of robust, infinite-dimensional generalizations of GBW and Log-Hilbert-Schmidt distances. Our approach also incorporates a learnable regularization parameter that enhances geometric stability in high-dimensional comparisons. Preliminary experiments reproducing benchmarks from the literature demonstrate the improved performance of our generalized metrics, particularly in scenarios involving comparisons between datasets of varying dimension and scale. This work lays a theoretical and computational foundation for advancing robust geometric methods in machine learning, statistical inference, and functional data analysis.