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.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
The generalization of the Bregman divergences on \(\mathrm{Sym}^{++}(n)\) to the infinite-dimensional setting will be presented in a separate paper.
References
Arsenin, V.I., Tikhonov, A.N.: Solutions of III-Posed Problems. Winston, Washington (1977)
Andruchow, E., Varela, A.: Non positively curved metric in the space of positive definite infinite matrices. Revista de la Union Mat. Argent. 48(1), 7–15 (2007)
Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Fast and simple calculus on tensors in the log-euclidean framework. In: Duncan, J.S., Gerig, G. (eds.) MICCAI 2005. LNCS, vol. 3749, pp. 115–122. Springer, Heidelberg (2005)
Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix An. App. 29(1), 328–347 (2007)
Barbaresco, F.: Information geometry of covariance matrix: Cartan-Siegel homogeneous bounded domains, Mostow/Berger fibration and Frechet median. In: Nielsen, F., Bhatia, R. (eds.) Matrix Information Geometry, pp. 199–255. Springer, Heidelberg (2013)
Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)
Bini, D.A., Iannazzo, B.: Computing the Karcher mean of symmetric positive definite matrices. Linear Algebra Appl. 438(4), 1700–1710 (2013)
Cherian, A., Sra, S., Banerjee, A., Papanikolopoulos, N.: Jensen-Bregman LogDet divergence with application to efficient similarity search for covariance matrices. TPAMI 35(9), 2161–2174 (2013)
Dryden, I.L., Koloydenko, A., Zhou, D.: Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Ann. Appl. Stat. 3, 1102–1123 (2009)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Mathematics and Its Applications, vol. 375. Springer, Netherlands (1996)
Formont, P., Ovarlez, J.-P., Pascal, F.: On the use of matrix information geometry for polarimetric SAR image classification. In: Nielsen, F., Bhatia, R. (eds.) Matrix Information Geometry, pp. 257–276. Springer, Heidelberg (2013)
Larotonda, G.: Geodesic Convexity, Symmetric Spaces and Hilbert-Schmidt Operators. Ph.D. thesis, Universidad Nacional de General Sarmiento, Buenos Aires, Argentina (2005)
Larotonda, G.: Nonpositive curvature: a geometrical approach to Hilbert-Schmidt operators. Differ. Geom. Appl. 25, 679–700 (2007)
Lawson, J., Lim, Y.: The least squares mean of positive Hilbert-Schmidt operators. J. Math. Anal. Appl. 403(2), 365–375 (2013)
Lawson, J.D., Lim, Y.: The geometric mean, matrices, metrics, and more. Am. Math. Monthly 108(9), 797–812 (2001)
Minh, H.Q., San Biagio, M., Murino, V.: Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces. In: Advances in Neural Information Processing Systems 27 (NIPS 2014), pp. 388–396 (2014)
Mostow, G.D.: Some new decomposition theorems for semi-simple groups. Memoirs Am. Math. Soc. 14, 31–54 (1955)
Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–66 (2006)
Petryshyn, W.V.: Direct and iterative methods for the solution of linear operator equations in Hilbert spaces. Trans. Am. Math. Soc. 105, 136–175 (1962)
Pigoli, D., Aston, J., Dryden, I.L., Secchi, P.: Distances and inference for covariance operators. Biometrika 101(2), 409–422 (2014)
Pigoli, D., Aston, J., Dryden, I.L., Secchi, P.: Permutation tests for comparison of covariance operators. In: Contributions in infinite-dimensional statistics and related topics, pp. 215–220. Società Editrice Esculapio (2014)
Qiu, A., Lee, A., Tan, M., Chung, M.K.: Manifold learning on brain functional networks in aging. Med. Image Anal. 20(1), 52–60 (2015)
Schölkopf, B., Smola, A., Müller, K.-R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10(5), 1299–1319 (1998)
Tosato, D., Spera, M., Cristani, M., Murino, V.: Characterizing humans on Riemannian manifolds. TPAMI 35(8), 1972–1984 (2013)
Tuzel, O., Porikli, F., Meer, P.: Pedestrian detection via classification on Riemannian manifolds. TPAMI 30(10), 1713–1727 (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-25040-3_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-25039-7
Online ISBN: 978-3-319-25040-3
eBook Packages: Computer ScienceComputer Science (R0)