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Generalized infinite dimensional Alpha-Procrustes based geometries

arXiv — stat.ML•Friday, November 14, 2025 at 5:00:00 AM
PositiveArtificial Intelligence
This work extends the Alpha-Procrustes family of Riemannian metrics for symmetric positive definite matrices by incorporating generalized versions of the Bures-Wasserstein, Log-Euclidean, and Wasserstein distances. The Alpha-Procrustes framework has unified many classical metrics in both finite and infinite-dimensional settings but previously lacked the structural components for these generalized forms. A formalism based on unitized Hilbert-Schmidt operators and an extended Mahalanobis norm is introduced, enabling robust infinite-dimensional generalizations of GBW and Log-Hilbert-Schmidt distances. The approach includes a learnable regularization parameter for enhanced geometric stability in high-dimensional comparisons. Preliminary experiments show improved performance of the generalized metrics.
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