Efficient and Guaranteed Rank Minimization by Atomic Decomposition

TL;DR

This paper introduces a new atomic decomposition-based algorithm for rank minimization that is both computationally efficient and comes with performance guarantees, suitable for large-scale problems.

Contribution

It extends the CoSaMP algorithm to matrix rank minimization using atomic decomposition, addressing scalability issues of previous semidefinite approaches.

Findings

Algorithm is computationally efficient for large matrices.

Provides theoretical performance guarantees.

Effectively handles large-scale rank minimization problems.

Abstract

Recht, Fazel, and Parrilo provided an analogy between rank minimization and $\ell_0$-norm minimization. Subject to the rank-restricted isometry property, nuclear norm minimization is a guaranteed algorithm for rank minimization. The resulting semidefinite formulation is a convex problem but in practice the algorithms for it do not scale well to large instances. Instead, we explore missing terms in the analogy and propose a new algorithm which is computationally efficient and also has a performance guarantee. The algorithm is based on the atomic decomposition of the matrix variable and extends the idea in the CoSaMP algorithm for $\ell_0$-norm minimization. Combined with the recent fast low rank approximation of matrices based on randomization, the proposed algorithm can efficiently handle large scale rank minimization problems.