Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow

This paper presents a new algorithm, termed <italic>truncated amplitude flow</italic> (TAF), to recover an unknown vector <inline-formula> <tex-math notation="LaTeX">$ {x}$ </tex-math></inline-formula> from a system of quadratic equations of the form <inline-formula> <tex-math notation="LaTeX">$y_{i}=|\langle {a}_{i}, {x}\rangle |^{2}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$ {a}_{i}$ </tex-math></inline-formula>’s are given random measurement vectors. This problem is known to be <italic>NP-hard</italic> 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 adapts the <italic>amplitude-based</italic> empirical loss function and proceeds in two stages. In the first stage, we introduce an <italic>orthogonality-promoting</italic> initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable <italic>truncated generalized gradient iterations</italic>, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors <inline-formula> <tex-math notation="LaTeX">$ {x}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${a}_{i}$ </tex-math></inline-formula>’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.