Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

TL;DR

This paper provides a comprehensive overview of recent advances in nonconvex optimization methods for low-rank matrix factorization, emphasizing statistical models, algorithmic strategies, and theoretical guarantees across various problems.

Contribution

It synthesizes recent theoretical and practical developments in nonconvex optimization for low-rank matrix problems, highlighting two main algorithmic approaches and their analysis.

Findings

Two-stage algorithms with initialization and refinement improve accuracy.

Landscape analysis enables initialization-free algorithms.

Statistical models are crucial for performance guarantees.

Abstract

Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.