Global convergence of the subgradient method for robust signal recovery

TL;DR

This paper establishes convergence guarantees for the subgradient method applied to nonsmooth, nonconvex robust signal recovery problems, including robust PCA and matrix sensing, under mild conditions.

Contribution

It develops a new convergence framework based on boundedness of subgradient trajectories for nonsmooth, nonconvex objectives, and verifies it for several robust recovery problems.

Findings

Subgradient method converges to critical points under mild boundedness conditions.

For robust PCA, the method avoids spurious critical points for almost all initializations.

Convergence to a global minimum is guaranteed in the rank-one robust PCA case.

Abstract

We study the subgradient method for factorized robust signal recovery problems, including robust PCA, robust phase retrieval, and robust matrix sensing. The resulting objectives are nonsmooth and nonconvex, and can have unbounded sublevel sets, so standard analyses based on descent and coercivity do not apply. For locally Lipschitz semialgebraic objectives, we develop a convergence framework that replaces these requirements with a boundedness condition on continuous-time subgradient trajectories. Under this condition and sufficiently small step sizes of order $1/k$, we show that iterates of the subgradient method remain bounded and the full sequence converges to a critical point. We then verify the required boundedness property for the three robust objectives by adapting existing trajectory analyses, assuming a mild nondegeneracy condition in the matrix sensing case. Finally, for rank-one symmetric robust PCA, we prove that for almost every initialization, the method cannot converge to spurious critical points; consequently, under the same step-size regime, it converges to a global minimum.