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International Conference on Geometric Science of Information

GSI 2015: Geometric Science of Information pp 30–38Cite as

Affine-Invariant Riemannian Distance Between Infinite-Dimensional Covariance Operators

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Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9389))

Abstract

This paper studies the affine-invariant Riemannian distance on the Riemann-Hilbert manifold of positive definite operators on a separable Hilbert space. This is the generalization of the Riemannian manifold of symmetric, positive definite matrices to the infinite-dimensional setting. In particular, in the case of covariance operators in a Reproducing Kernel Hilbert Space (RKHS), we provide a closed form solution, expressed via the corresponding Gram matrices.

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Notes

  1. 1.

    The generalization of the Bregman divergences on \(\mathrm{Sym}^{++}(n)\) to the infinite-dimensional setting will be presented in a separate paper.

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Author information

Authors and Affiliations

  1. Pattern Analysis and Computer Vision (PAVIS), Istituto Italiano di Tecnologia (IIT), Genova, Italy

    Hà Quang Minh

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  1. Hà Quang Minh
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Correspondence to Hà Quang Minh .

Editor information

Editors and Affiliations

  1. Bâtiment Alan Turing, CS35003, École Polytechnique, Palaiseau, France

    Frank Nielsen

  2. Thales Land\& Air Systems, Limours, France

    Frédéric Barbaresco

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Minh, H.Q. (2015). Affine-Invariant Riemannian Distance Between Infinite-Dimensional Covariance Operators. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2015. Lecture Notes in Computer Science(), vol 9389. Springer, Cham. https://doi.org/10.1007/978-3-319-25040-3_4

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  • DOI: https://doi.org/10.1007/978-3-319-25040-3_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-25039-7

  • Online ISBN: 978-3-319-25040-3

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