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This paper deals with finding an $n$-dimensional solution $x$ to a system ofquadratic equations of the form $y_i=|\langle{a}_i,x\rangle|^2$ for $1\le i \lem$, which is also known as phase retrieval and is NP-hard in general. We putforth a novel procedure for minimizing the amplitude-based least-squaresempirical loss, that starts with a weighted maximal correlation initializationobtainable with a few power or Lanczos iterations, followed by successiverefinements based upon a sequence of iteratively reweighted (generalized)gradient iterations. The two (both the initialization and gradient flow) stagesdistinguish themselves from prior contributions by the inclusion of a fresh(re)weighting regularization technique. The overall algorithm is conceptuallysimple, numerically scalable, and easy-to-implement. For certain randommeasurement models, the novel procedure is shown capable of finding the truesolution $x$ in time proportional to reading the data $\{(a_i;y_i)\}_{1\le i\le m}$. This holds with high probability and without extra assumption on thesignal $x$ to be recovered, provided that the number $m$ of equations is someconstant $c>0$ times the number $n$ of unknowns in the signal vector, namely,$m>cn$. Empirically, the upshots of this contribution are: i) (almost) $100\%$perfect signal recovery in the high-dimensional (say e.g., $n\ge 2,000$) regimegiven only an information-theoretic limit number of noiseless equations,namely, $m=2n-1$ in the real-valued Gaussian case; and, ii) (nearly) optimalstatistical accuracy in the presence of additive noise of bounded support.Finally, substantial numerical tests using both synthetic data and real imagescorroborate markedly improved signal recovery performance and computationalefficiency of our novel procedure relative to state-of-the-art approaches.

Solving Almost all Systems of Random Quadratic Equations