Scalable Algorithms for Tractable Schatten Quasi-Norm Minimization

TL;DR

This paper introduces scalable algorithms for Schatten quasi-norm minimization that are faster and more accurate than existing methods, by defining new tractable quasi-norms and designing efficient optimization algorithms.

Contribution

The paper defines two new tractable Schatten quasi-norms and develops efficient algorithms with convergence guarantees for large-scale matrix completion tasks.

Findings

Algorithms are significantly faster than state-of-the-art methods.

Proposed methods achieve higher accuracy in matrix completion.

The algorithms have proven global convergence and better performance guarantees.

Abstract

The Schatten-p quasi-norm $(0<p<1)$ is usually used to replace the standard nuclear norm in order to approximate the rank function more accurately. However, existing Schatten-p quasi-norm minimization algorithms involve singular value decomposition (SVD) or eigenvalue decomposition (EVD) in each iteration, and thus may become very slow and impractical for large-scale problems. In this paper, we first define two tractable Schatten quasi-norms, i.e., the Frobenius/nuclear hybrid and bi-nuclear quasi-norms, and then prove that they are in essence the Schatten-2/3 and 1/2 quasi-norms, respectively, which lead to the design of very efficient algorithms that only need to update two much smaller factor matrices. We also design two efficient proximal alternating linearized minimization algorithms for solving representative matrix completion problems. Finally, we provide the global convergence and performance guarantees for our algorithms, which have better convergence properties than existing algorithms. Experimental results on synthetic and real-world data show that our algorithms are more accurate than the state-of-the-art methods, and are orders of magnitude faster.