Affine-invariant Riemannian Distance Between Infinite-dimensional Covariance Operators

28/10/2015
Auteurs : Minh Ha Quang
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14318
DOI : http://dx.doi.org/10.1007/978-3-319-25040-3_4You do not have permission to access embedded form.

Résumé

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.

Affine-invariant Riemannian Distance Between Infinite-dimensional Covariance Operators

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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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Affine-invariant Riemannian distance between infinite-dimensional covariance operators H`a Quang Minh Istituto Italiano di Tecnologia, ITALY Affine-invariant Riemannian distance between infinite-dimensional covariance operators – p.1/52 From finite to infinite dimensions Affine-invariant Riemannian distance between infinite-dimensional covariance operators – p.2/52 Outline 1. Review of finite-dimensional setting: Affine-invariant Riemannian metric on the manifold of symmetric positive definite matrices 2. Infinite-dimensional generalization: Riemann-Hilbert manifold of positive definite unitized Hilbert-Schmidt operators 3. Affine-invariant Riemannian distance between Reproducing Kernel Hilbert Spaces (RKHS) covariance operators Affine-invariant Riemannian distance between infinite-dimensional covariance operators – p.3/52 Positive definite matrices Sym++ (n) = symmetric, positive definite n × n matrices Have been studied extensively mathematically Numerous practical applications Brain imaging (Arsigny et al 2005, Dryden et al 2009, Qiu et al 2015) Computer vision: object detection (Tuzel et al 2008, Tosato et al 2013), image retrieval (Cherian et al 2013), visual recognition (Jayasumana et al 2015) Radar signal processing: Barbaresco (2013), Formont et al 2013 Machine learning: kernel learning (Kulis et al 2009) Affine-invariant Riemannian distance between infinite-dimensional covariance operators – p.4/52 Positive definite matrices Sym++ (n) = symmetric, positive definite n × n matrices Differentiable manifold viewpoint Tangent space TP (Sym++ )(n) ∼= Sym(n) = vector space of symmetric matrices Affine-invariant Riemannian metric: on TP (Sym++ (n)) A, B P = P−1/2 AP−1/2 , P−1/2 BP−1/2 F = tr[P−1 AP−1 B] with the Frobenius inner product A, B F = tr(AT B) Affine-invariant Riemannian distance between infinite-dimensional covariance operators – p.5/52 Positive definite matrices Riemannian metric: on TP (Sym++ (n)) = Sym(n) A, B P = P−1/2 AP−1/2 , P−1/2 BP−1/2 F with the Frobenius inner product A, B F = tr(AT B) Affine-invariance CACT , CBCT CPCT = A, B P for any matrix C ∈ GL(n) Siegel (1943), Mostow (1955), Pennec et al 2006, Bhatia 2007, Moakher and Zéraï 2011, Bini and Iannazzo 2013 Affine-invariant Riemannian distance between infinite-dimensional covariance operators – p.6/52 Positive definite matrices Geodesically complete, with nonpositive curvature Geodesic joining P, Q ∈ Sym++ (n) γPQ(t) = P1/2 (P−1/2 QP−1/2 )t P1/2 The exponential map ExpP : TP (Sym++ (n)) → Sym++ (n) ExpP (V ) = P1/2 exp(P−1/2 V P−1/2 )P1/2 is defined on all of TP (Sym++ (n) Affine-invariant Riemannian distance between infinite-dimensional covariance operators – p.7/52 Positive definite matrices Riemannian distance daiE(A, B) = || log(A−1/2 BA−1/2 ||F where log(A) is the principal logarithm of A A