A Riemannian geometry with complete geodesics for the set of positive semidefinite matrices of fixed rank

We present a homogeneous space geometry for the manifold of symmetric positive semidefinite matrices of fixed rank. The total space is the general linear group endowed with its natural rightinvariant metric, and the metric on the homogeneous space is chosen such that the quotient space is the image of a Riemannian submersion from the total space. As a result, we obtain complete geodesics that are the image of certain geodesics on the general linear group. We derive in addition an efficient closed-form expression for these geodesics. Furthermore, we isometrically embed the abstract quotient space into the space of real matrices. This allows us to interpret the vector fields, the metric and the geodesics in terms of concrete matrices. Finally, we compare our geometry with some other geometries in the literature. In particular, we show how other geodesics and their approximations relate to ours.

[1]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[2]  K. Nomizu,et al.  Foundations of Differential Geometry , 1963 .

[3]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[4]  J. Cheeger,et al.  Comparison theorems in Riemannian geometry , 1975 .

[5]  Begnaud Francis Hildebrand,et al.  Introduction to numerical analysis: 2nd edition , 1987 .

[6]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[7]  R. Mahony.  Optimization algorithms on homogeneous spaces: with application in linear systems theory , 1995, Journal and proceedings of the Royal Society of New South Wales.

[8]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[9]  Uwe Helmke,et al.  Critical points of matrix least squares distance functions , 1995 .

[10]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[11]  John M. Lee.  Riemannian Manifolds: An Introduction to Curvature , 1997 .

[12]  Thilo Penzl,et al.  Numerical solution of generalized Lyapunov equations , 1998, Adv. Comput. Math..

[13]  Katya Scheinberg,et al.  Efficient SVM Training Using Low-Rank Kernel Representations , 2002, J. Mach. Learn. Res..

[14]  G. Stewart.  Matrix Algorithms, Volume II: Eigensystems , 2001 .

[15]  M. Postnikov,et al.  Geometry VI: Riemannian Geometry , 2001 .

[16]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[17]  John M. Lee.  Introduction to Smooth Manifolds , 2002 .

[18]  P. Priouret,et al.  Newton's method on Riemannian manifolds: covariant alpha theory , 2002, math/0209096.

[19]  R. Adler,et al.  Newton's method on Riemannian manifolds and a geometric model for the human spine , 2002 .

[20]  Orizon Pereira Ferreira,et al.  Kantorovich's Theorem on Newton's Method in Riemannian Manifolds , 2002, J. Complex..

[21]  Michael J. Todd,et al.  On the Riemannian Geometry Defined by Self-Concordant Barriers and Interior-Point Methods , 2002, Found. Comput. Math..

[22]  Alain Trouvé,et al.  The metric spaces, Euler equations, and normal geodesic image motions of computational anatomy , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[23]  Nello Cristianini,et al.  Learning the Kernel Matrix with Semidefinite Programming , 2002, J. Mach. Learn. Res..

[24]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[25]  John B. Moore,et al.  A Newton-like method for solving rank constrained linear matrix inequalities , 2006, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[26]  P. Absil,et al.  Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation , 2004 .

[27]  Xavier Pennec,et al.  A Riemannian Framework for Tensor Computing , 2005, International Journal of Computer Vision.

[28]  Shotaro Akaho,et al.  Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold , 2005, Neurocomputing.

[29]  S.T. Smith,et al.  Covariance, subspace, and intrinsic Crame/spl acute/r-Rao bounds , 2005, IEEE Transactions on Signal Processing.

[30]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[31]  Chong Li,et al.  Newton's method on Riemannian manifolds: Smale's point estimate theory under the γ-condition , 2006 .

[32]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[33]  Robert E. Mahony,et al.  Optimization Algorithms on Matrix Manifolds , 2007 .

[34]  Yaguang Yang.  Globally Convergent Optimization Algorithms on Riemannian Manifolds: Uniform Framework for Unconstrained and Constrained Optimization , 2007 .

[35]  Frank Chongwoo Park,et al.  Geometric Direct Search Algorithms for Image Registration , 2007, IEEE Transactions on Image Processing.

[36]  Othmar Koch,et al.  Dynamical Low-Rank Approximation , 2007, SIAM J. Matrix Anal. Appl..

[37]  Pierre-Antoine Absil,et al.  Trust-Region Methods on Riemannian Manifolds , 2007, Found. Comput. Math..

[38]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[39]  Felipe Alvarez,et al.  A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds , 2008, Found. Comput. Math..

[40]  R. Sepulchre,et al.  Geometric distance and mean for positive semi-definite matrices of fixed rank , 2008 .

[41]  Bart Vandereycken,et al.  Embedded geometry of the set of symmetric positive semidefinite matrices of fixed rank , 2009, 2009 IEEE/SP 15th Workshop on Statistical Signal Processing.

[42]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2009, Found. Comput. Math..

[43]  Sabine Van Huffel,et al.  A Geometric Newton Method for Oja's Vector Field , 2008, Neural Computation.

[44]  Silvere Bonnabel,et al.  From subspace learning to distance learning: A geometrical optimization approach , 2009, 2009 IEEE/SP 15th Workshop on Statistical Signal Processing.

[45]  Rodolphe Sepulchre,et al.  Adaptive filtering for estimation of a low-rank positive semidefinite matrix , 2010 .

[46]  Francis R. Bach,et al.  Low-Rank Optimization on the Cone of Positive Semidefinite Matrices , 2008, SIAM J. Optim..

[47]  Stefan Vandewalle,et al.  A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations , 2010, SIAM J. Matrix Anal. Appl..

[48]  Silvère Bonnabel,et al.  Géométrie des matrices positives semi-définies de rang fixé : un peu de théorie et beaucoup d'applications , 2010 .

[49]  Feng Ye.  Semi-Riemannian Geometry , 2011 .

[50]  Yonina C. Eldar,et al.  Uniqueness conditions for low-rank matrix recovery , 2011, Optical Engineering + Applications.

[51]  G. Larotonda,et al.  The left invariant metric in the general linear group , 2011, 1109.0520.

[52]  Silvere Bonnabel,et al.  Regression on Fixed-Rank Positive Semidefinite Matrices: A Riemannian Approach , 2010, J. Mach. Learn. Res..

[53]  Silvere Bonnabel,et al.  Rank-preserving geometric means of positive semi-definite matrices , 2010, 1007.5494.