Orthogonal Rank-One Matrix Pursuit for Low Rank Matrix Completion

TL;DR

This paper introduces an efficient orthogonal rank-one pursuit algorithm for low rank matrix completion, demonstrating superior scalability and convergence, with strong empirical results on large datasets like Netflix.

Contribution

It extends orthogonal matching pursuit to matrices, introduces a weight updating rule for efficiency, and proves linear convergence, improving over existing methods.

Findings

Achieves linear convergence rate.

More efficient than state-of-the-art algorithms.

Performs well on large-scale datasets like Netflix.

Abstract

In this paper, we propose an efficient and scalable low rank matrix completion algorithm. The key idea is to extend orthogonal matching pursuit method from the vector case to the matrix case. We further propose an economic version of our algorithm by introducing a novel weight updating rule to reduce the time and storage complexity. Both versions are computationally inexpensive for each matrix pursuit iteration, and find satisfactory results in a few iterations. Another advantage of our proposed algorithm is that it has only one tunable parameter, which is the rank. It is easy to understand and to use by the user. This becomes especially important in large-scale learning problems. In addition, we rigorously show that both versions achieve a linear convergence rate, which is significantly better than the previous known results. We also empirically compare the proposed algorithms with several state-of-the-art matrix completion algorithms on many real-world datasets, including the large-scale recommendation dataset Netflix as well as the MovieLens datasets. Numerical results show that our proposed algorithm is more efficient than competing algorithms while achieving similar or better prediction performance.