Scalable Robust Matrix Factorization with Nonconvex Loss

TL;DR

This paper introduces a scalable, robust matrix factorization method using nonconvex loss and an efficient optimization algorithm, outperforming existing approaches in accuracy and scalability especially with sparse data.

Contribution

It proposes a novel nonconvex loss for robustness, along with a scalable MM-based optimization leveraging data sparsity and dual optimization techniques.

Findings

Outperforms state-of-the-art in accuracy

Demonstrates superior scalability

Guarantees convergence to a critical point

Abstract

Robust matrix factorization (RMF), which uses the $\ell_1$-loss, often outperforms standard matrix factorization using the $\ell_2$-loss, particularly when outliers are present. The state-of-the-art RMF solver is the RMF-MM algorithm, which, however, cannot utilize data sparsity. Moreover, sometimes even the (convex) $\ell_1$-loss is not robust enough. In this paper, we propose the use of nonconvex loss to enhance robustness. To address the resultant difficult optimization problem, we use majorization-minimization (MM) optimization and propose a new MM surrogate. To improve scalability, we exploit data sparsity and optimize the surrogate via its dual with the accelerated proximal gradient algorithm. The resultant algorithm has low time and space complexities and is guaranteed to converge to a critical point. Extensive experiments demonstrate its superiority over the state-of-the-art in terms of both accuracy and scalability.