NDLI: Fixed-rank matrix factorizations and Riemannian low-rank optimization

Content Provider	Springer Nature Link
Author	Mishra, Bamdev Meyer, Gilles Bonnabel, Silvère Sepulchre, Rodolphe
Copyright Year	2013
Abstract	Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems, and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss the usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with state-of-the-art algorithms and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix.
Starting Page	591
Ending Page	621
Page Count	31
File Format	PDF
ISSN	09434062
Journal	Computational Statistics
Volume Number	29
Issue Number	3-4
e-ISSN	16139658
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2013-11-12
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Riemannian quotient geometry Riemannian trust-region Steepest descent Low-rank matrix completion Linear regression Statistics Probability and Statistics in Computer Science Probability Theory and Stochastic Processes Economic Theory
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty Computational Mathematics

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