Fast Rank One Alternating Minimization Algorithm for Phase Retrieval

TL;DR

This paper introduces a fast, alternating gradient descent algorithm for phase retrieval that splits variables to solve a quadratic bi-variate problem, leading to faster convergence than existing methods.

Contribution

The paper proposes a novel variable-splitting strategy and an alternating gradient descent algorithm with proven convergence for phase retrieval, improving speed over traditional Wirtinger flow methods.

Findings

Faster convergence than Wirtinger flow in numerical experiments.

Requires fewer iterations to reach the same accuracy.

Convergence is guaranteed for any initialization.

Abstract

The phase retrieval problem is a fundamental problem in many fields, which is appealing for investigation. It is to recover the signal vector $\tilde{x}\in\mathbb{C}^d$ from a set of $N$ measurements $b_n=|f^*_n\tilde{x}|^2,\ n=1,\cdots, N$, where $\{f_n\}_{n=1}^N$ forms a frame of $\mathbb{C}^d$. %It is generally a non-convex minimization problem, which is NP-hard. Existing algorithms usually use a least squares fitting to the measurements, yielding a quartic polynomial minimization. In this paper, we employ a new strategy by splitting the variables, and we solve a bi-variate optimization problem that is quadratic in each of the variables. An alternating gradient descent algorithm is proposed, and its convergence for any initialization is provided. Since a larger step size is allowed due to the smaller Hessian, the alternating gradient descent algorithm converges faster than the gradient descent algorithm (known as the Wirtinger flow algorithm) applied to the quartic objective without splitting the variables. Numerical results illustrate that our proposed algorithm needs less iterations than Wirtinger flow to achieve the same accuracy.