Conic geometric optimization on the manifold of positive definite matrices

SIAM Journal on Optimization

Volumn 25, Issue 1, 2015, Pages 713-739

  Sra, Suvrit ^a   Hosseini, Reshad ^b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF TEHRAN   (Iran)

Author keywords

Conic fixed point theory; Geodesic convexity; Geometric optimization; Log nonexpansive; Manifold optimization; Riemannian bfgs; Thompson metric; Vector transport

Indexed keywords

ALGORITHMS; COST FUNCTIONS; FIXED POINT ARITHMETIC; FUNCTIONS; GEOMETRY; GLOBAL OPTIMIZATION; MAXIMUM LIKELIHOOD; MAXIMUM LIKELIHOOD ESTIMATION; NORMAL DISTRIBUTION;

FIXED POINT THEORY; GEODESIC CONVEXITIES; GEOMETRIC OPTIMIZATION; NONEXPANSIVE; RIEMANNIAN BFGS; THOMPSON; VECTOR TRANSPORT;

OPTIMIZATION;

EID: 84983354772     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/140978168     Document Type: Conference Paper

References (52)

1. P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, Princeton, NJ, 2009.
2. T. Ando, Concavity of certain maps of positive definite matrices and applications to Hadamard products, Linear Algebra Appl., 26 (1979), pp. 203-221.
3. T. Ando, C.-K. Li, and R. Mathias, Geometric means, Linear Algebra Appl., 385 (2004), pp. 305-334.
4. M. Arnaudon, F. Barbaresco, and L. Yang, Riemannian medians and means with applications to radar signal processing, IEEE J. Selected Topics Signal Process., 7 (2013), pp. 595-604.
5. R. Bhatia, Matrix Analysis, Springer, New York, 1997.
6. R. Bhatia, Positive Definite Matrices, Princeton University Press, Princeton, NJ, 2007.
7. R. Bhatia and R. L. Karandikar, Monotonicity of the matrix geometric mean, Math. Ann., 353 (2012), pp. 1453-1467.
8. D. A. Bini and B. Iannazzo, Computing the Karcher mean of symmetric positive definite matrices, Linear Algebra Appl., 438 (2013), pp. 1700-1710.
9. G. Blekherman, P. A. Parrilo, and R. R. Thomas, eds., Semidefinite Optimization and Convex Algebraic Geometry, SIAM, Philadelphia, 2013.
10. N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre, Manopt, a MATLAB toolbox for optimization on manifolds, J. Machine Learning Res., 15 (2014), pp. 1455-1459.
11. S. Boyd, S.-J. Kim, L. Vandenberghe, and A. Hassibi, A tutorial on geometric programming, Optimization Engrg., 8 (2007), pp. 67-127.
12. M. R. Bridson and A. Haeflinger, Metric Spaces of Non-Positive Curvature, Springer, New York, 1999.
13. S. Cambanis, S. Huang, and G. Simons, On the theory of elliptically contoured distributions, J. Multivariate Anal., 11 (1981), pp. 368-385.
14. Z. Chebbi and M. Moahker, Means of Hermitian positive-definite matrices based on the log-determinant α-divergence function, Linear Algebra Appl., 436 (2012), pp. 1872-1889.
15. Y. Chen, A. Wiesel, and A. O. Hero, Robust shrinkage estimation of high-dimensional covariance matrices, IEEE Trans. Signal Process., 59 (2011), pp. 4097-4107.
16. G. Cheng, H. Salehian, and B. C. Vemuri, Efficient recursive algorithms for computing the mean diffusion tensor and applications to DTI segmentation, in Proceedings of the European Conference on Computer Vision (ECCV), Vol. 7, Springer, New York, 2012, pp. 390-401.
17. G. Cheng and B. C. Vemuri, A novel dynamic system in the space of SPD matrices with applications to appearance tracking, SIAM J. Imaging Sci., 6 (2013), pp. 592-615.
18. A. Cherian, S. Sra, A. Banerjee, and N. Papanikolopoulos, Jensen-Bregman Log Det divergence for efficient similarity computations on positive definite tensors, IEEE Trans. Pattern Anal. Machine Intell., 35 (2013), pp. 2161-2174.
19. M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl., 10 (1975), pp. 285-290.
20. M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., s1/37 (1962), pp. 74-79.
21. A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, Chapman and Hall/CRC, Boca Raton, FL, 1999.
22. L. Gurvits and A. Samorodnitsky, A deterministic algorithm for approximating mixed discriminant and mixed volume, and a combinatorial corollary, Discrete Comput. Geom., 27 (2002), pp. 531-550.
23. G. H. Hardy, J. E. Littlewood, and G. Pólya, Some simple inequalities satisfied by convex functions, Messenger Math., 58 (1929), pp. 145-152.
24. F. Hiai and D. Petz, Riemannian metrics on positive definite matrices related to means. II, Linear Algebra Appl., 436 (2012), pp. 2117-2136.
25. B. Jeuris, R. Vandebril, and B. Vandereycken, A survey and comparison of contemporary algorithms for computing the matrix geometric mean, Electron. Trans. Numer. Anal., 39 (2012), pp. 379-402.
26. S. Kotz, K.-T. Fang, and K. W. Ng, Symmetric Multivariate and Related Distributions, Chapman and Hall, London, 1990.
27. J. T. Kent and D. E. Tyler, Redescending M-estimates of multivariate location and scatter, Ann. Statist., 19 (1991), pp. 2102-2119.
28. S. Kotz, N. L. Johnson, and D. W. Boyd, Series representations of distributions of quadratic forms in normal variables. I. Central case, Ann. Math. Statist., 38 (1967), pp. 823-837.
29. K. Kraus, General state changes in quantum theory, Ann. Phys., 64 (1971), pp. 311-335.
30. F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246 (1980), pp. 205- 224.
31. H. Lee and Y. Lim, Invariant metrics, contractions and nonlinear matrix equations, Nonlinearity, 21 (2008), pp. 857-878.
32. B. Lemmens and R. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge University Press, Cambridge, UK, 2012.
33. Y. Lim and M. Pálfia, Matrix power means and the Karcher mean, J. Funct. Anal., 262 (2012), pp. 1498-1514.
34. C. F. Van Loan, The ubiquitous Kronecker product, J. Comput. Appl. Math., 123 (2000), pp. 85-100.
35. J. S. Matharu and J. S. Aujla, Some inequalities for unitarily invariant norms, Linear Algebra Appl., 436 (2012), pp. 1623-1631.
36. M. Moakher, A differential geometric approach to the geometric mean of symmetric positivedefinite matrices, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 735-747.
37. R. J. Muirhead, Aspects of Multivariate Statistical Theory, John Wiley, New York, 1982.
38. Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, SIAM, Philadelphia, 1994.
39. C. P. Niculescu, Convexity according to the geometric mean, Math. Inequalities Appl., 3 (2000), pp. 155-167.
40. C. Niculesu and L. E. Persson, Convex Functions and Their Applications: A Contemporary Approach, Science & Business 13, Springer, New York, 2006.
41. F. Nielsen and R. Bhatia, eds., Matrix Information Geometry, Springer, New York, 2013.
42. E. Ollila, D.E. Tyler, V. Koivunen, and H. V. Poor, Complex elliptically symmetric distributions: Survey, new results and applications, IEEE Trans. Signal Process., 60 (2011), pp. 5597-5625.
43. A. Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature, European Mathematical Society, Zürich, Switzerland, 2005.
44. C. Qi, K. A. Gallivan, and P.-A. Absil, Riemannian BFGS algorithm with applications, in Recent Advances in Optimization and Its Applications in Engineering, Springer, Berlin, Heidelberg, 2010, pp. 183-192.
45. T. Rapcsák, Geodesic convexity in nonlinear optimization, J. Optim. Theory Appl., 69 (1991), pp. 169-183.
46. W. Ring and B. Wirth, Optimization methods on Riemannian manifolds and their application to shape space, SIAM J. Optim., 22 (2012), pp. 596-627.
47. R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, New York, 1998.
48. S. Sra, Positive definite matrices and the S-divergence, arXiv preprint, arXiv: 1110.1773v4, http://arxiv.org/abs/1110.1773v4, 2013.
49. A. Wiesel, Geodesic convexity and covariance estimation, IEEE Trans. Signal Process., 60 (2012), pp. 6182-89.
50. A. L. Yuille and A. Rangarajan, The concave-convex procedure, Neural Comput., 15 (2003), pp. 915-936.
51. T. Zhang, A. Wiesel, and S. Greco, Multivariate generalized Gaussian distribution: Convexity and graphical models, IEEE Trans. Signal Process., 60 (2013), pp. 5597-5625.
52. H. Zhu, H. Zhang, J.G. Ibrahim, and B.S. Peterson, Statistical analysis of diffusion tensors in diffusion-weighted magnetic resonance imaging data, J. Amer. Statist. Assoc., 102 (2007), pp. 1085-1102.