Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares

TL;DR

This paper introduces a fast alternating least squares algorithm that unifies nuclear-norm regularization and matrix factorization methods for efficient large-scale matrix completion, outperforming existing approaches.

Contribution

It presents a novel algorithm combining two popular matrix completion methods, with software implementations in R and Spark for large-scale applications.

Findings

Outperforms existing matrix completion algorithms

Efficient for large matrices using distributed computing

Provides software package 'softImpute' and Spark version

Abstract

The matrix-completion problem has attracted a lot of attention, largely as a result of the celebrated Netflix competition. Two popular approaches for solving the problem are nuclear-norm-regularized matrix approximation (Candes and Tao, 2009, Mazumder, Hastie and Tibshirani, 2010), and maximum-margin matrix factorization (Srebro, Rennie and Jaakkola, 2005). These two procedures are in some cases solving equivalent problems, but with quite different algorithms. In this article we bring the two approaches together, leading to an efficient algorithm for large matrix factorization and completion that outperforms both of these. We develop a software package "softImpute" in R for implementing our approaches, and a distributed version for very large matrices using the "Spark" cluster programming environment.