Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation

TL;DR

This paper reviews recent advances in low-rank matrix estimation from incomplete data, highlighting convex and nonconvex methods, their theoretical guarantees, and applications with additional structural properties.

Contribution

It provides a unified overview of computationally efficient algorithms for low-rank matrix estimation, including their theoretical performance guarantees and handling of additional structures.

Findings

Convex relaxations like nuclear norm minimization are statistically optimal.

Nonconvex procedures such as projected gradient descent often achieve global optimality efficiently.

Recent theoretical results characterize the performance of these algorithms under various conditions.

Abstract

Low-rank modeling plays a pivotal role in signal processing and machine learning, with applications ranging from collaborative filtering, video surveillance, medical imaging, to dimensionality reduction and adaptive filtering. Many modern high-dimensional data and interactions thereof can be modeled as lying approximately in a low-dimensional subspace or manifold, possibly with additional structures, and its proper exploitations lead to significant reduction of costs in sensing, computation and storage. In recent years, there is a plethora of progress in understanding how to exploit low-rank structures using computationally efficient procedures in a provable manner, including both convex and nonconvex approaches. On one side, convex relaxations such as nuclear norm minimization often lead to statistically optimal procedures for estimating low-rank matrices, where first-order methods are developed to address the computational challenges; on the other side, there is emerging evidence that properly designed nonconvex procedures, such as projected gradient descent, often provide globally optimal solutions with a much lower computational cost in many problems. This survey article will provide a unified overview of these recent advances on low-rank matrix estimation from incomplete measurements. Attention is paid to rigorous characterization of the performance of these algorithms, and to problems where the low-rank matrix have additional structural properties that require new algorithmic designs and theoretical analysis.