Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems

Abstract

We consider the fundamental problem of solving quadratic systems of equations in n variables, where y i D jha i ; xij 2 , i D 1; : : : ; m, and x 2 R n is unknown. We propose a novel method, which starts with an initial guess computed by means of a spectral method and proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach [13]. There are several key distinguishing features, most notably a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e., in time proportional to reading the data fa i g and fy i g as soon as the ratio m=n between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems in which we only have y i jha i ; xij 2 and prove that our algorithms achieve a statistical accuracy, which is nearly unimprovable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size-hence the title of this paper. For instance, we demonstrate empirically that the computational cost of our algorithm is about four times that of solving a least squares problem of the same size.

Sparse superresolution phase retrieval from phase-coded noisy intensity patterns

“…14,25 Methodologically, these algorithms are developed for the Poissonian likelihood criterion, i.e., for Poissonian noisy observations. Simulation experiments confirm that these algorithm works precisely provided nearly noiseless observations.…”

Detection with phaseless measurements

“…For example, PhaseLift requires O(N ) measurements [13] for recovery with high probability, but involves solving a semidefinite program of size N × N . The Truncated Wirtinger Flow method requires O(N log N ) measurements and yields a solution with relative accuracy in MN 2 log 1/ flops [15]. In contrast, our proposed method requires O(M 2 N ) flops for the least-squares solution (which can be performed offline if the measurement vectors are known ahead of time), and M multiplications plus summations to calculate the statistic for each set of measurements.…”

“…For sparse signals, there are phaseless recovery algorithms based on convex relaxations [6] as well as nonlinear formulations [7]. Other methods are based on Wirtinger Flow descent [14,15], alternating minimization [16] and generalized approximate message passing (GAMP) [17]. A method with low complexity which requires a quadratic number of measurements was described in [18], though a specific structure is assumed for the measurement vectors.…”

On gradient descent algorithm for generalized phase retrieval problem

“…The convergence of NCG and BFGS for general non-convex problems is not guaranteed [27], and we cannot prove that for the generalized phase retrieval problem at present. Besides, the sampling complexity may be further reduced to O(n) by borrowing the idea of truncated Wirtinger flow in literate [28]. It is also worth investigating whether the performance of numerical algorithm can be further improved by exploring the structures of the signal, such as sparsity [29].…”