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

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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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Notes

  1. There is a typo in the corresponding result in Theorem 4 in [26], with the constant in the Fisher–Rao distance being 1/2. The correct constant is \(1/\sqrt{2}\) as stated here.

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Correspondence to Hà Quang Minh.

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Communicated by Frank Nielsen.

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Minh, H.Q. Fisher–Rao geometry of equivalent Gaussian measures on infinite-dimensional Hilbert spaces. Info. Geo. (2024). https://doi.org/10.1007/s41884-024-00137-0

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