Global Optimality of Local Search for Low Rank Matrix Recovery

TL;DR

This paper proves that local search methods like stochastic gradient descent can reliably find global solutions in low-rank matrix recovery problems, even with noise, due to the absence of spurious local minima.

Contribution

It establishes the global optimality landscape for non-convex low-rank matrix recovery, showing no spurious local minima and providing convergence guarantees.

Findings

No spurious local minima in the non-convex formulation.

All local minima are close to the global optimum with noisy data.

Polynomial time convergence of stochastic gradient descent from random initialization.

Abstract

We show that there are no spurious local minima in the non-convex factorized parametrization of low-rank matrix recovery from incoherent linear measurements. With noisy measurements we show all local minima are very close to a global optimum. Together with a curvature bound at saddle points, this yields a polynomial time global convergence guarantee for stochastic gradient descent {\em from random initialization}.