Neural Information Processing Systems http://nips.cc/

Here, the functionf() is applied toAx in a component-wise manner. The above model arises in many applications of signal processing [13, 10, 41], communications [56, 9, 25], and machine learning [48, 40].

Our algorithm is simple and can be obtained via a natural combination of the classical alternating minimization approach for phase retrieval, with the CoSaMP algorithm for sparse recovery.

This model is motivated by applications to channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is difficult or impossible to acquire. We show that when a is random and m is sufficiently large, x can be efficiently recovered up to a global phase using a combination of spectral initialization and generalized gradient descent. The main challenge is coping with dependencies in the measurement operator; we overcome this challenge by using ideas from decoupling theory, suprema of chaos processes and the restricted isometry property of random circulant matrices, and recent analysis for alternating minimizing methods.

Recovering a signal via a quadratic system of equations has gained intensive attention recently.

Neural Information Processing Systems http://nips.cc/

In many machine learning applications one optimizes a non-convex loss function; this is often achieved using simple descending algorithms such as gradient descent or its stochastic variations.

It turns out that the "spikiness" (or conversely "flatness") of the spectrum of the sensing