Learning on the compact Stiefel manifold by a cayley-transform-based pseudo-retraction map

The present research takes its moves from previous contributions by the present authors on two topics, namely, neural learning on differentiable manifolds by manifold retractions and averaging over differentiable manifolds. Learning on differentiable manifolds is a general theory that allows a neural system that insists on curved smooth spaces to adapt its parameters without violating the constraints on the geometry of the parameter spaces. In particular, the present contribution focuses on learning on the compact Stiefel manifold by manifold retraction with application to averaging `tall-skinny' matrices and generalizes some contributions recently appeared in the scientific literature about such a topic.

[1]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[2]  F. Tompkins,et al.  Bayesian Filtering on the Stiefel Manifold , 2007, 2007 2nd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing.

[3]  Peter Meer,et al.  Nonlinear Mean Shift over Riemannian Manifolds , 2009, International Journal of Computer Vision.

[4]  Paul Van Dooren,et al.  Optimization over the Stiefel manifold , 2007 .

[5]  Elena Celledoni,et al.  On the Implementation of Lie Group Methods on the Stiefel Manifold , 2003, Numerical Algorithms.

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

[7]  Simone Fiori,et al.  Averaging over the Lie group of optical systems transference matrices , 2011 .

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

[9]  Rama Chellappa,et al.  Statistical Computations on Grassmann and Stiefel Manifolds for Image and Video-Based Recognition , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[10]  Simone G. O. Fiori,et al.  Lie-group-type neural system learning by manifold retractions , 2008, Neural Networks.

[11]  Simone G. O. Fiori,et al.  Descent methods for optimization on homogeneous manifolds , 2008, Math. Comput. Simul..

[12]  Benjamin Berkels,et al.  An SL(2) Invariant Shape Median , 2010, Journal of Mathematical Imaging and Vision.

[13]  Hasan Ertan Ceting.  Intrinsic Mean Shift for Clustering on Stiefel and Grassmann Manifolds , 2009 .

[14]  Simone G. O. Fiori,et al.  Extended Hamiltonian Learning on Riemannian Manifolds: Theoretical Aspects , 2011, IEEE Transactions on Neural Networks.

[15]  Nilima Nigam,et al.  Geometric integration on spheres and some interesting applications , 2003 .

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

[17]  Andrzej Cichocki,et al.  Adaptive blind signal and image processing , 2002 .

[18]  Feng Yin,et al.  An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB=C , 2008 .

[19]  Toshihisa Tanaka,et al.  An Algorithm to Compute Averages on Matrix Lie Groups , 2009, IEEE Transactions on Signal Processing.

[20]  Marc Arnaudon,et al.  Medians and Means in Riemannian Geometry: Existence, Uniqueness and Computation , 2011, 1111.3120.

[21]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[22]  Simone G. O. Fiori,et al.  Solving Minimal-Distance Problems over the Manifold of Real-Symplectic Matrices , 2011, SIAM J. Matrix Anal. Appl..

[23]  Simone G. O. Fiori,et al.  Extended Hamiltonian Learning on Riemannian Manifolds: Numerical Aspects , 2012, IEEE Transactions on Neural Networks and Learning Systems.