Riemannian-Gradient-Based Learning on the Complex Matrix-Hypersphere

This brief tackles the problem of learning over the complex-valued matrix-hypersphere Sn,pα(C). The developed learning theory is formulated in terms of Riemannian-gradient-based optimization of a regular criterion function and is implemented by a geodesic-stepping method. The stepping method is equipped with a geodesic-search sub-algorithm to compute the optimal learning stepsize at any step. Numerical results show the effectiveness of the developed learning method and of its implementation.

[1]  Qin Lin,et al.  A unified algorithm for principal and minor components extraction , 1998, Neural Networks.

[2]  Giuseppe Caire,et al.  An iterative water-filling algorithm for maximum weighted sum-rate of Gaussian MIMO-BC , 2006, IEEE Journal on Selected Areas in Communications.

[3]  Simone G. O. Fiori,et al.  Learning by Natural Gradient on Noncompact Matrix-Type Pseudo-Riemannian Manifolds , 2010, IEEE Transactions on Neural Networks.

[4]  António Guedes de Oliveira.  On the Adjugate of a Matrix , 2007, Am. Math. Mon..

[5]  Hanif D. Sherali,et al.  On the Maximum Weighted Sum-Rate of MIMO Gaussian Broadcast Channels , 2008, 2008 IEEE International Conference on Communications.

[6]  Elena Celledoni,et al.  Norges Teknisk-naturvitenskapelige Universitet Neural Learning by Geometric Integration of Reduced 'rigid-body' Equations Neural Learning by Geometric Integration of Reduced 'rigid-body' Equations , 2022 .

[7]  Paul de Kerret,et al.  An algorithm for maximizing a quotient of two Hermitian form determinants with different exponents , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[8]  Simone G. O. Fiori,et al.  Geodesic-based and projection-based neural blind deconvolution algorithms , 2008, Signal Process..

[9]  Lei Xu,et al.  Theories for unsupervised learning: PCA and its nonlinear extensions , 1994, Proceedings of 1994 IEEE International Conference on Neural Networks (ICNN'94).

[10]  Simone G. O. Fiori,et al.  Blind separation of circularly distributed sources by neural extended APEX algorithm , 2000, Neurocomputing.

[11]  Shlomo Shamai,et al.  On the achievable throughput of a multiantenna Gaussian broadcast channel , 2003, IEEE Transactions on Information Theory.

[12]  Shun-ichi Amari,et al.  Geometrical methods in neural networks and learning , 2005, Neurocomputing.

[13]  Peter Tino,et al.  IEEE Transactions on Neural Networks , 2009 .

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

[15]  Simone G. O. Fiori,et al.  Nonlinear Complex-Valued Extensions of Hebbian Learning: An Essay , 2005, Neural Computation.

[16]  Wolfgang Utschick,et al.  Efficient Weighted Sum Rate Maximization With Linear Precoding , 2010, IEEE Transactions on Signal Processing.

[17]  Simone G. O. Fiori,et al.  Quasi-Geodesic Neural Learning Algorithms Over the Orthogonal Group: A Tutorial , 2005, J. Mach. Learn. Res..

[18]  D. Gabay.  Minimizing a differentiable function over a differential manifold , 1982 .

[19]  Shun-ichi Amari,et al.  Unified stabilization approach to principal and minor components extraction algorithms , 2001, Neural Networks.

[20]  Shun-ichi Amari,et al.  Editorial : Special issue on “ Geometrical Methods in Neural Networks and Learning ” , .

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

[22]  Simone G. O. Fiori,et al.  A Study on Neural Learning on Manifold Foliations: The Case of the Lie Group SU(3) , 2008, Neural Computation.

[23]  Simone G. O. Fiori,et al.  A theory for learning based on rigid bodies dynamics , 2002, IEEE Trans. Neural Networks.

[24]  Yuhua Zhu,et al.  Optimal dimension reduction for image retrieval with correlation metrics , 2009, 2009 International Joint Conference on Neural Networks.

[25]  Akira Hirose,et al.  Complex-Valued Neural Networks: Theories and Applications , 2003 .

[26]  D. Luenberger.  The Gradient Projection Method Along Geodesics , 1972 .

[27]  Simone G. O. Fiori,et al.  On vector averaging over the unit hypersphere , 2009, Digit. Signal Process..

[28]  Simone G. O. Fiori,et al.  Formulation and integration of learning differential equations on the stiefel manifold , 2005, IEEE Transactions on Neural Networks.

[29]  Akira Hirose,et al.  Complex-Valued Neural Networks , 2006, Studies in Computational Intelligence.

[30]  Simone G. O. Fiori,et al.  Fast fixed-point neural blind-deconvolution algorithm , 2004, IEEE Transactions on Neural Networks.

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