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Affine-Invariant Riemannian Distance Between Infinite-Dimensional Covariance Operators

Part of the Lecture Notes in Computer Science book series (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.

Keywords

  • Covariance Operator
  • Separable Hilbert Space
  • Reproduce Kernel Hilbert Space
  • Scalar Perturbation
  • Positive Definite Matrice

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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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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