Large-Scale Convex Minimization with a Low-Rank Constraint

TL;DR

This paper introduces an efficient greedy algorithm for large-scale convex matrix minimization with low-rank constraints, providing approximation guarantees and scalable to real-world applications like matrix completion.

Contribution

It proposes a novel greedy algorithm with theoretical guarantees for low-rank convex minimization, scalable to large matrices in practical applications.

Findings

Algorithm scales linearly with matrix size

Provides formal approximation guarantees

Effective in matrix completion and low-rank approximation

Abstract

We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation guarantees. Each iteration of the algorithm involves (approximately) finding the left and right singular vectors corresponding to the largest singular value of a certain matrix, which can be calculated in linear time. This leads to an algorithm which can scale to large matrices arising in several applications such as matrix completion for collaborative filtering and robust low rank matrix approximation.