Inertial Proximal ADMM for Separable Multi-Block Convex Optimizations and Compressive Affine Phase Retrieval

TL;DR

This paper introduces an inertial proximal ADMM method with proven global convergence for solving multi-block convex optimization problems and applies it to develop an algorithm for sparse affine phase retrieval from noisy measurements.

Contribution

It proposes a novel inertial proximal ADMM for multi-block convex problems and connects affine phase retrieval with convex optimization for sparse signal recovery.

Findings

The inertial proximal ADMM converges globally under mild conditions.

The proposed algorithm effectively recovers sparse signals from noisy affine quadratic measurements.

Numerical results demonstrate satisfactory performance in affine phase retrieval.

Abstract

Separable multi-block convex optimization problem appears in many mathematical and engineering fields. In the first part of this paper, we propose an inertial proximal ADMM to solve a linearly constrained separable multi-block convex optimization problem, and we show that the proposed inertial proximal ADMM has global convergence under mild assumptions on the regularization matrices. Affine phase retrieval arises in holography, data separation and phaseless sampling, and it is also considered as a nonhomogeneous version of phase retrieval that has received considerable attention in recent years. Inspired by convex relaxation of vector sparsity and matrix rank in compressive sensing and by phase lifting in phase retrieval, in the second part of this paper, we introduce a compressive affine phase retrieval via lifting approach to connect affine phase retrieval with multi-block convex optimization, and then based on the proposed inertial proximal ADMM for multi-block convex optimization, we propose an algorithm to recover sparse real signals from their (noisy) affine quadratic measurements. Our numerical simulations show that the proposed algorithm has satisfactory performance for affine phase retrieval of sparse real signals.