Fixed-rank matrix factorizations and Riemannian low-rank optimization

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. B. Mishra (B) · G. Meyer · R. Sepulchre Department of Electrical Engineering andComputer Science, University of Liège, 4000Liege, Belgium e-mail: [email protected] G. Meyer e-mail: [email protected] R. Sepulchre Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK e-mail: [email protected] S. Bonnabel Robotics Center, Mines ParisTech, Boulevard Saint-Michel, 60, 75272 Paris, France e-mail: [email protected]