A new metric on the manifold of kernel matrices with application to matrix geometric means

Advances in Neural Information Processing Systems

Volumn 1, Issue , 2012, Pages 144-152

  Sra, Suvrit ^a  

^a MAX PLANCK INSTITUTE FOR INTELLIGENT SYSTEMS   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

DISTANCE FUNCTIONS; MEASURING DISTANCES; NON-EUCLIDEAN DISTANCE; NON-EUCLIDEAN GEOMETRY; RIEMANNIAN METRICS; SCIENTIFIC DISCIPLINE; SQUARED DISTANCES; SYMMETRIC POSITIVE DEFINITE;

GEOMETRY;

MATRIX ALGEBRA;

EID: 84877770427     PISSN: 10495258     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (33)

1. H. Lee and Y. Lim. Invariant metrics, contractions and nonlinear matrix equations. Nonlinearity, 21: 857-878, 2008.
2. Z. Chebbi and M. Moahker. Means of hermitian positive-definite matrices based on the log-determinant α-divergence function. Linear Algebra and its Applications, 436:1872-1889, 2012.
3. P. Bougerol. Kalman Filtering with Random Coefficients and Contractions. SIAM J. Control Optim., 31 (4):942-959, 1993.
4. A. Cherian, S. Sra, A. Banerjee, and N. Papanikolopoulos. Efficient Similarity Search for Covariance Matrices via the Jensen-Bregman LogDet Divergence. In International Conference on Computer Vision (ICCV), Nov. 2011.
5. F. Porikli, O. Tuzel, and P. Meer. Covariance Tracking using Model Update Based on Lie Algebra. In IEEE CVPR, 2006.
6. L. T. Skovgaard. A Riemannian Geometry of the Multivariate Normal Model. Scandinavian Journal of Statistics, 11(4):211-223, 1984.
7. D. Petz. Quantum Information Theory and Quantum Statistics. Springer, 2008.
8. I. Dryden, A. Koloydenko, and D. Zhou. Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging. Annals of Applied Statistics, 3(3):1102-1123, 2009.
9. H. Zhu, H. Zhang, J. G. Ibrahim, and B. S. Peterson. Statistical Analysis of Diffusion Tensors in Diffusion-Weighted Magnetic Resonance Imaging Data. Journal of the American Statistical Association, 102(480): 1085-1102, 2007.
10. F. Hiai and D. Petz. Riemannian metrics on positive definite matrices related to means. Linear Algebra and its Applications, 430:3105-3130, 2009.
11. R. Bhatia. Positive Definite Matrices. Princeton University Press, 2007.
12. M. R. Bridson and A. Haeflinger. Metric Spaces of Non-Positive Curvature. Springer, 1999.
13. A. Terras. Harmonic Analysis on Symmetric Spaces and Applications, volume II. Springer, 1988.
14. Yu. Nesterov and A. Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. SIAM, 1987.
15. A. Ben-Tal and A. Nemirovksii. Lectures on modern convex optimization: Analysis, algorithms, and engineering applications. SIAM, 2001.
16. Yu. Nesterov and M. J. Todd. On the riemannian geometry defined for self-concordant barriers and interior point methods. Found. Comput. Math., 2:333-361, 2002.
17. S. Helgason. Geometric Analysis on Symmetric Spaces. Number 39 in Mathematical Surveys and Monographs. AMS, second edition, 2008.
18. H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors. Handbook of Semidefinite Programming: Theory, Algorithms, and Applications. Kluwer Academic, 2000.
19. S. Sra. Positive definite matrices and the Symmetric Stein Divergence. arXiv: 1110.1773, October 2012.
20. C. Berg, J. P. R. Christensen, and P. Ressel. Harmonic analysis on semigroups: theory of positive definite and related functions, volume 100 of GTM. Springer, 1984.
21. R. Bhatia. Matrix Analysis. Springer, 1997.
22. R. Bellman. Introduction to Matrix Analysis. SIAM, second edition, 1970.
23. M. Harandi, C. Sanderson, R. Hartley, and B. Lovell. Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach. In European Conference on Computer Vision (ECCV), 2012.
24. M. Cuturi, K. Fukumizu, and J. P. Vert. Semigroup kernels on measures. JMLR, 6:1169-1198, 2005.
25. R. J. Muirhead. Aspects of multivariate statistical theory. Wiley Interscience, 1982.
26. J. Faraut and A. Korányi. Analysis on Symmetric Cones. Clarendon Press, 1994.
27. M.-F. Bru. Wishart Processes. J. Theoretical Probability, 4(4), 1991.
28. S. G. Gindikin. Invariant generalized functions in homogeneous domains. Functional Analysis and its Applications, 9:50-52, 1975.
29. A. Cherian, S. Sra, A. Banerjee, and N. Papanikolopoulos. Jensen-Bregman LogDet Divergence with Application to Efficient Similarity Search for Covariance Matrices. IEEE TPAMI, 2012. Submitted.
30. T. Ando. Concavity of certain maps on positive definite matrices and applications to hadamard products. Linear Algebra and its Applications, 26(0):203-241, 1979.
31. R. Bhatia and J. A. R. Holbrook. Riemannian geometry and matrix geometric means. Linear Algebra Appl., 413:594-618, 2006.
32. M. Moakher. A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. (SIMAX), 26:735-747, 2005.
33. D. A. Bini and B. Iannazzo. Computing the Karcher mean of symmetric positive definite matrices. Linear Algebra and its Applications, Oct. 2011. Available online.