Solving Systems of Random Quadratic Equations via Truncated Amplitude Flow

Abstract

This paper presents a new algorithm, termed truncated amplitude flow (TAF), to recover an unknown vector x from a system of quadratic equations of the form yi = | ai, x | 2 , where ai's are given random measurement vectors. This problem is known to be 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 amplitude-based empirical loss function, and proceeds in two stages. In the first stage, we introduce an orthogonality-promoting initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors x and ai'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.

Convolutional Phase Retrieval via Gradient Descent

“…measurements. Those methods include the truncated spectral method [8], null initialization [56] and orthogonality-promoting initialization [9]. For the simplicity of analysis, here we only consider Algorithm 1 for the convolutional model.…”

“…provably recovers the ground truth, with near-optimal sample complexity m ≥ Ω(n log n). The subsequent work [8,9,35] further reduced the sample complexity to m ≥ Ω(n) by using different nonconvex objectives and truncation techniques. In particular, recent work by [9,35] studied a nonsmooth objective that is similar to ours (1.4) with weighting b = 1.…”

“…where τ > 0 is the stepsize. Indeed, ∂ ∂z f (z) can be interpreted as the subgradient of f (z) in the real case; this method can be seen as a variant of amplitude flow [9].…”

Almost optimal phaseless compressed sensing with sublinear decoding time

“…However, PhaseLift is computationally very expensive, making it impractical. Non-convex formulations such as Wirtinger flows [CLS15,CC15] or Truncated Amplitude Flow [WGE16] have been developed; the algorithms they use, however, are complicated with many parameters. Very recently, a convex approach called PhaseMax [BR16,GS16] has been suggested and analyzed; the best known analysis of PhaseMax [HV16] shows that PhaseMax succeeds in finding the underlying vector x under optimal sample complexity.…”

Randomized Block Frank–Wolfe for Convergent Large-Scale Learning

“…Typically, solving a constrained linear optimization is considerably easier than finding the aforementioned projections per iteration. The resulting savings benefit diverse large-scale learning tasks, including matrix completion [6], multi-class classification [7], image reconstruction [7], structural support vector machines (SVMs) [8], particle filtering [9], sparse phase retrieval [10], [11], and scheduling electric vehicle (EV) charging [12].…”