The Riemannian Mean of Positive Matrices

Recent work in the study of the geometric mean of positive definite matrices has seen the coming together of several subjects: matrix analysis, operator theory, differential geometry (Riemannian and Finsler), probability and numerical analysis. At the same time the range of its applications has grown from physics and electrical engineering (the two areas in which the subject had its beginnings) to include radar data processing, medical imaging, elasticity, statistics and machine learning. This article, based on my talk at the Indo-French Seminar on Matrix Information Geometries, is a partial view of the arena from the perspective of matrix analysis. There has been striking progress on one of the problems raised in that talk, and I report on that as well. Apertinentreferenceforthetheoryofmatrixmeansis[8],Chaps.4and6.General facts on matrix analysis used here can be found in [6].

[1]  T. Andô,et al.  Means of positive linear operators , 1980 .

[2]  Horacio Porta,et al.  Convexity of the geodesic distance on spaces of positive operators , 1994 .

[3]  Tomohiro Hayashi.  A note on the Jensen inequality for self-adjoint operators , 2009, 1203.1154.

[4]  Y. Lim,et al.  Monotonic properties of the least squares mean , 2010, 1007.4792.

[5]  R. Bhatia.  On the exponential metric increasing property , 2003 .

[6]  R. Subramanian,et al.  Inequalities between means of positive operators , 1978, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  R. Bhatia,et al.  Riemannian geometry and matrix geometric means , 2006 .

[8]  R. Bhatia.  Positive Definite Matrices , 2007 .

[9]  Noboru Nakamura,et al.  Geometric Means of Positive Operators , 2009 .

[10]  R. Bhatia,et al.  Noncommutative geometric means , 2006 .

[11]  PRODUCTS OF BLOCK TOEPLITZ OPERATORS , 1996, math/9605225.

[12]  Frank Nielsen,et al.  Medians and means in Finsler geometry , 2010, LMS J. Comput. Math..

[13]  Frédéric Barbaresco,et al.  Interactions between Symmetric Cone and Information Geometries: Bruhat-Tits and Siegel Spaces Models for High Resolution Autoregressive Doppler Imagery , 2009, ETVC.

[14]  Y. Lim,et al.  Matrix power means and the Karcher mean , 2012 .

[15]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[16]  Dario Bini,et al.  Computing the Karcher mean of symmetric positive definite matrices , 2013 .

[17]  Jimmie D. Lawson,et al.  The Geometric Mean, Matrices, Metrics, and More , 2001, Am. Math. Mon..

[18]  Dan Rutherford,et al.  Hyponormality of block Toeplitz operators , 2006 .

[19]  H. Karcher.  Riemannian center of mass and mollifier smoothing , 1977 .

[20]  Takeaki Yamazaki.  An elementary proof of arithmetic–geometric mean inequality of the weighted Riemannian mean of positive definite matrices , 2013 .

[21]  Federico Poloni,et al.  An effective matrix geometric mean satisfying the Ando-Li-Mathias properties , 2010, Math. Comput..

[22]  R. Karandikar,et al.  Monotonicity of the matrix geometric mean , 2012 .

[23]  Karl-Theodor Sturm,et al.  Probability Measures on Metric Spaces of Nonpositive Curvature , 2003 .

[24]  Miklós Pálfia,et al.  A Multivariable Extension of Two-Variable Matrix Means , 2011, SIAM J. Matrix Anal. Appl..

[25]  Takeaki Yamazaki.  The Riemannian mean and matrix inequalities related to the Ando-Hiai inequality and chaotic order , 2012 .

[26]  Maher Moakher,et al.  A Differential Geometric Approach to the Geometric Mean of Symmetric Positive-Definite Matrices , 2005, SIAM J. Matrix Anal. Appl..

[27]  Maher Moakher,et al.  To appear in: SIAM J. MATRIX ANAL. APPL. MEANS AND AVERAGING IN THE GROUP OF ROTATIONS∗ , 2002 .

[28]  T. Andô.  Concavity of certain maps on positive definite matrices and applications to Hadamard products , 1979 .

[29]  Maher Moakher.  On the Averaging of Symmetric Positive-Definite Tensors , 2006 .

[30]  W. Pusz,et al.  Functional calculus for sesquilinear forms and the purification map , 1975 .

[31]  Chi-Kwong Li.  Geometric Means , 2003 .