How Many Samples is a Good Initial Point Worth in Low-rank Matrix Recovery?

TL;DR

This paper investigates how the quality of an initial guess influences the number of samples needed for successful low-rank matrix recovery, showing that better initializations significantly reduce data requirements.

Contribution

It quantifies the relationship between initial guess quality and sample complexity, providing sharp thresholds for avoiding spurious local minima based on the initial point.

Findings

Better initial guesses lead to fewer samples needed.

A linear improvement in initial guess quality results in a constant factor reduction in data requirements.

The analysis provides sharp thresholds for sample complexity depending on initialization quality.

Abstract

Given a sufficiently large amount of labeled data, the non-convex low-rank matrix recovery problem contains no spurious local minima, so a local optimization algorithm is guaranteed to converge to a global minimum starting from any initial guess. However, the actual amount of data needed by this theoretical guarantee is very pessimistic, as it must prevent spurious local minima from existing anywhere, including at adversarial locations. In contrast, prior work based on good initial guesses have more realistic data requirements, because they allow spurious local minima to exist outside of a neighborhood of the solution. In this paper, we quantify the relationship between the quality of the initial guess and the corresponding reduction in data requirements. Using the restricted isometry constant as a surrogate for sample complexity, we compute a sharp threshold number of samples needed to prevent each specific point on the optimization landscape from becoming a spurious local minimum. Optimizing the threshold over regions of the landscape, we see that for initial points around the ground truth, a linear improvement in the quality of the initial guess amounts to a constant factor improvement in the sample complexity.