Optimal Sample Complexity of Subgradient Descent for Amplitude Flow via Non-Lipschitz Matrix Concentration

TL;DR

This paper proves that subgradient descent can recover signals from phaseless measurements with optimal sample complexity using novel concentration inequalities for a discontinuous matrix operator.

Contribution

It introduces a new uniform matrix concentration inequality for a non-Lipschitz operator, enabling analysis of subgradient descent in amplitude flow problems.

Findings

Subgradient descent converges linearly with optimal sample complexity.

Discontinuous matrix operators can satisfy uniform concentration inequalities.

The techniques extend to traditional inverse problems beyond neural network priors.

Abstract

We consider the problem of recovering a real-valued $n$-dimensional signal from $m$ phaseless, linear measurements and analyze the amplitude-based non-smooth least squares objective. We establish local convergence of subgradient descent with optimal sample complexity based on the uniform concentration of a random, discontinuous matrix-valued operator arising from the objective's gradient dynamics. While common techniques to establish uniform concentration of random functions exploit Lipschitz continuity, we prove that the discontinuous matrix-valued operator satisfies a uniform matrix concentration inequality when the measurement vectors are Gaussian as soon as $m = \Omega(n)$ with high probability. We then show that satisfaction of this inequality is sufficient for subgradient descent with proper initialization to converge linearly to the true solution up to the global sign ambiguity. As a consequence, this guarantees local convergence for Gaussian measurements at optimal sample complexity. The concentration methods in the present work have previously been used to establish recovery guarantees for a variety of inverse problems under generative neural network priors. This paper demonstrates the applicability of these techniques to more traditional inverse problems and serves as a pedagogical introduction to those results.