arXiv:2511.09801v2 Announce Type: replace 
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.

تم تقديم تقدم حديث في عائلة مقاييس ريمان Alpha-Procrustes، مما يوسع تطبيقها ليشمل نسخًا عامة من مسافات Bures-Wasserstein وLog-Euclidean وWasserstein. يعتمد هذا التطور على مشغلات Hilbert-Schmidt الموحدة ومعيار Mahalanobis الموسع لإنشاء تعميمات لا نهائية قوية، مما يعزز الاستقرار الهندسي في المقارنات عالية الأبعاد.

Se han introducido avances recientes en la familia de métricas Riemannianas Alpha-Procrustes, ampliando su aplicación a versiones generalizadas de las distancias Bures-Wasserstein, Log-Euclidiana y Wasserstein. Este desarrollo se basa en operadores de Hilbert-Schmidt unificados y una norma de Mahalanobis extendida para crear generalizaciones infinitas robustas, mejorando la estabilidad geométrica en comparaciones de alta dimensión.

Des avancées récentes dans la famille des métriques riemanniennes Alpha-Procrustes ont été introduites, étendant son application à des versions généralisées des distances Bures-Wasserstein, Log-Euclidienne et Wasserstein. Ce développement s'appuie sur des opérateurs de Hilbert-Schmidt unifiés et une norme de Mahalanobis étendue pour créer des généralisations infinies robustes, améliorant la stabilité géométrique dans les comparaisons de haute dimension.

Recent advancements in the Alpha-Procrustes family of Riemannian metrics have been introduced, extending its application to generalized versions of Bures-Wasserstein, Log-Euclidean, and Wasserstein distances. This development leverages unitized Hilbert-Schmidt operators and an extended Mahalanobis norm to create robust infinite-dimensional generalizations, enhancing geometric stability in high-dimensional comparisons.

Generalized infinite dimensional Alpha-Procrustes based geometries

arXiv:2601.08184v1 Announce Type: cross 
Abstract: Finite-time central limit theorem (CLT) rates play a central role in modern machine learning (ML). In this paper, we study CLT rates for multivariate dependent data in Wasserstein-$p$ ($\mathcal W_p$) distance, for general $p\ge 1$. We focus on two fundamental dependence structures that commonly arise in ML: locally dependent sequences and geometrically ergodic Markov chains. In both settings, we establish the \textit{first optimal} $\mathcal O(n^{-1/2})$ rate in $\mathcal W_1$, as well as the first $\mathcal W_p$ ($p\ge 2$) CLT rates under mild moment assumptions, substantially improving the best previously known bounds in these dependent-data regimes. As an application of our optimal $\mathcal W_1$ rate for locally dependent sequences, we further obtain the first optimal $\mathcal W_1$--CLT rate for multivariate $U$-statistics.
  On the technical side, we derive a tractable auxiliary bound for $\mathcal W_1$ Gaussian approximation errors that is well suited to studying dependent data. For Markov chains, we further prove that the regeneration time of the split chain associated with a geometrically ergodic chain has a geometric tail without assuming strong aperiodicity or other restrictive conditions. These tools may be of independent interests and enable our optimal $\mathcal W_1$ rates and underpin our $\mathcal W_p$ ($p\ge 2$) results.

أظهرت دراسة حديثة معدلات مثالية لنظرية الحد المركزي (CLT) في الزمن المحدد للبيانات متعددة المتغيرات المعتمدة في مسافة فاسرشتاين-$p$، مع التركيز على تسلسلات معتمدة محليًا وسلاسل ماركوف ذات تآزر هندسي. تكشف النتائج عن أول معدل مثالي $	ext{O}(n^{-1/2})$ في $	ext{W}_1$ وتحسينات كبيرة لمعدلات $	ext{W}_p$ تحت افتراضات لحظية خفيفة.

Un estudio reciente ha establecido tasas óptimas del teorema central del límite (TCL) en tiempo finito para datos multivariantes dependientes en la distancia de Wasserstein-$p$, centrándose en secuencias localmente dependientes y cadenas de Markov ergódicas geométricamente. Los hallazgos revelan la primera tasa óptima de $	ext{O}(n^{-1/2})$ en $	ext{W}_1$ y mejoras significativas para las tasas de $	ext{W}_p$ bajo supuestos de momento leves.

Une étude récente a établi des taux optimaux du théorème central limite (TCL) en temps fini pour des données multivariées dépendantes dans la distance de Wasserstein-$p$, en se concentrant sur des séquences localement dépendantes et des chaînes de Markov géométriquement ergodiques. Les résultats révèlent le premier taux optimal de $	ext{O}(n^{-1/2})$ dans $	ext{W}_1$ et des améliorations significatives pour les taux de $	ext{W}_p$ sous des hypothèses de moment légères.

A recent study has established optimal finite-time central limit theorem (CLT) rates for multivariate dependent data in Wasserstein-$p$ distance, focusing on locally dependent sequences and geometrically ergodic Markov chains. The findings reveal the first optimal $	ext{O}(n^{-1/2})$ rate in $	ext{W}_1$ and significant improvements for $	ext{W}_p$ rates under mild moment assumptions.

Generalized infinite dimensional Alpha-Procrustes based geometries

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