Fisher-Rao geometry of equivalent Gaussian measures on infinite-dimensional Hilbert spaces

2023/10/16 by Quang, Minh Ha · 1 citation
Mathematics · #28C20 #47B65 #60G15 #Differential Geometry (math.DG) #FOS: Mathematics #Morphological variations and asymmetry #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2310.10182

openalex publication_date 2023/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work presents an explicit description of the Fisher-Rao Riemannian metric on the Hilbert manifold of equivalent centered Gaussian measures on an infinite-dimensional Hilbert space. We show that the corresponding quantities from the finite-dimensional setting of Gaussian densities on Euclidean space, including the Riemannian metric, Levi-Civita connection, curvature, geodesic curve, and Riemannian distance, when properly formulated, directly generalize to this setting. Furthermore, we discuss the connection with the Riemannian geometry of positive definite unitized Hilbert-Schmidt operators on Hilbert space, which can be viewed as a regularized version of the current setting.

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