Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow

TL;DR

This paper introduces the truncated amplitude flow (TAF) algorithm for solving quadratic systems, proving its efficiency and accuracy in recovering unknown vectors with high probability, and demonstrating superior empirical performance over existing methods.

Contribution

The paper develops a novel two-stage amplitude-based algorithm with a new initialization and truncation scheme, providing theoretical guarantees and improved empirical results for solving quadratic equations.

Findings

TAF recovers solutions exactly with high probability when equations are sufficient in number.

The algorithm's complexity grows linearly with problem size.

TAF outperforms existing Wirtinger flow variants in numerical tests.

Abstract

This paper presents a new algorithm, termed \emph{truncated amplitude flow} (TAF), to recover an unknown vector $\bm{x}$ from a system of quadratic equations of the form $y_i=|\langle\bm{a}_i,\bm{x}\rangle|^2$, where $\bm{a}_i$'s are given random measurement vectors. This problem is known to be \emph{NP-hard} in general. We prove that as soon as the number of equations is on the order of the number of unknowns, TAF recovers the solution exactly (up to a global unimodular constant) with high probability and complexity growing linearly with both the number of unknowns and the number of equations. Our TAF approach adopts the \emph{amplitude-based} empirical loss function, and proceeds in two stages. In the first stage, we introduce an \emph{orthogonality-promoting} initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable \emph{truncated generalized gradient iterations}, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors $\bm{x}$ and $\bm{a}_i$'s are real-valued, our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version. Numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations. Furthermore, even under the same initialization, the proposed amplitude-based refinement outperforms existing Wirtinger flow variants, corroborating the superior performance of TAF over state-of-the-art algorithms.