Year: 2025
Type: preprint
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.
Source: arXiv (Cornell University)
Institutions: Centre National de la Recherche Scientifique, Institut national de recherche en sciences et technologies du numérique, Inserm, Collège de France, Université Paris Sciences et Lettres, Laboratoire d'Informatique, Signaux et Systèmes de Sophia Antipolis, Centre Interdisciplinaire de Recherche en Biologie, The University of Sydney, RIKEN, RIKEN Center for Advanced Intelligence Project, Indian Institute of Technology Bombay, Université Côte d'Azur
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4416237697
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