Fixed-rank matrix factorizations and Riemannian low-rank optimization

Bamdev Mishra (), Gilles Meyer (), Silvère Bonnabel () and Rodolphe Sepulchre ()

Computational Statistics, 2014, vol. 29, issue 3, 621 pages

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. Copyright Springer-Verlag Berlin Heidelberg 2014

Keywords: Riemannian quotient geometry; Riemannian trust-region; Steepest descent; Low-rank matrix completion; Linear regression (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s00180-013-0464-z

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