Low-complexity implementation of convex optimization-based phase retrieval

TL;DR

This paper explores low-complexity convex optimization algorithms for phase retrieval, demonstrating their efficiency and minimal performance loss in optical imaging and communications applications.

Contribution

It introduces and compares three low-complexity algorithms—projected gradient, Nesterov, and ADMM—for convex phase retrieval, highlighting ADMM's faster convergence.

Findings

All methods have quadratic complexity in unknown parameters.

Estimated penalties are less than 0.6 dB in applications.

ADMM converges in fewer iterations than other methods.

Abstract

Phase retrieval has important applications in optical imaging, communications and sensing. Lifting the dimensionality of the problem allows phase retrieval to be approximated as a convex optimization problem in a higher-dimensional space. Convex optimization-based phase retrieval has been shown to yield high accuracy, yet its low-complexity implementation has not been explored. In this paper, we study three fundamental approaches for its low-complexity implementation: the projected gradient method, the Nesterov accelerated gradient method, and the alternating direction method of multipliers (ADMM) method. We derive the corresponding estimation algorithms and evaluate their complexities. We compare their performance in the application area of direct-detection mode-division multiplexing. We demonstrate that they yield negligible estimation penalties (less than 0.2 dB for transmitter processing and less than 0.6 dB for receiver equalization) while yielding low computational cost, as their implementation complexities all scale quadratically in the number of unknown parameters. Among the three methods, ADMM achieves convergence after the smallest number of iterations.