Quantum mechanics

Gaussian

Low-rank approximation

Pure mathematics

Materials science

Social psychology

Applied mathematics

Rank (graph theory)

Eigenvalues and eigenvectors

Mathematical optimization

Bounded function

Matrix (chemical analysis)

Tensor (intrinsic definition)

Physics

Psychology

Mathematics

Relaxation (psychology)

Mathematical analysis

Geometry

Composite material

Combinatorics

Matrix norm

Convex optimization

Matrix completion

Regular polygon

SIAM Journal on Optimization

This paper studies noisy low-rank matrix completion: given partial and noisy entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practice. However, the theoretical support of this approach is still far from optimal in the noisy setting, falling short of explaining its empirical success. We make progress towards demystifying the practical efficacy of convex relaxation vis-à-vis random noise. When the rank and the condition number of the unknown matrix are bounded by a constant, we demonstrate that the convex programming approach achieves near-optimal estimation errors---in terms of the Euclidean loss, the entrywise loss, and the spectral norm loss---for a wide range of noise levels. All of this is enabled by bridging convex relaxation with the nonconvex Burer--Monteiro approach, a seemingly distinct algorithmic paradigm that is provably robust against noise. More specifically, we show that an approximate critical point of the nonconvex formulation serves as an extremely tight approximation of the convex solution, thus allowing us to transfer the desired statistical guarantees of the nonconvex approach to its convex counterpart.

Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization