CUED Publications database

Fixed-rank matrix factorizations and Riemannian low-rank optimization

Mishra, B and Meyer, G and Bonnabel, S and Sepulchre, R (2014) Fixed-rank matrix factorizations and Riemannian low-rank optimization. Computational Statistics, 29. pp. 591-621. ISSN 0943-4062

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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. © 2013 Springer-Verlag Berlin Heidelberg.

Item Type: Article
Divisions: Div F > Control
Depositing User: Unnamed user with email
Date Deposited: 17 Jul 2017 18:57
Last Modified: 09 Sep 2021 02:43
DOI: 10.1007/s00180-013-0464-z