A Convergent Gradient Descent Algorithm for Rank Minimization and Semidefinite Programming from Random Linear Measurements

TL;DR

This paper introduces a scalable gradient descent algorithm that efficiently solves rank minimization and semidefinite programming problems from a limited number of random measurements, with guaranteed linear convergence.

Contribution

The paper presents a novel gradient descent method for nonconvex optimization in rank minimization and semidefinite programming with provable convergence guarantees from random measurements.

Findings

Linear convergence to the global optimum.

Requires $O(r^3 ppa^2 n \u221alog n)$ measurements.

Effective for positive semidefinite matrices of rank r.

Abstract

We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs. With $O(r^3 \kappa^2 n \log n)$ random measurements of a positive semidefinite $n \times n$ matrix of rank $r$ and condition number $\kappa$, our method is guaranteed to converge linearly to the global optimum.