Time: Tuesday, March 8, 2016

Speaker: Dr. Yi-Kai Liu (NIST)

Title: Phase Retrieval using Structured Measurements

Abstract: Phase retrieval is the task of learning an unknown n-dimensional vector x (over the real or complex numbers) from quadratic measurements. That is, one is given measurements of the squared inner products y_i = |a_i^T x|^2, for i = 1,2,...,m, where the vectors a_i are chosen by the observer. Such measurements arise in a variety of applications, including coherent diffractive imaging, and quantum state tomography. There is a natural approach to solving the phase retrieval problem, by means of a convex relaxation called PhaseLift. Furthermore, PhaseLift is known to perform well (with provable recovery guarantees) when the measurement vectors a_i are chosen independently at random from a Gaussian distribution. Unfortunately, these Gaussian measurements are difficult to implement in real experiments. In this talk, I will show theoretical results on the performance of PhaseLift with different kinds of structured measurements. These include Bernoulli measurements (sampled from the uniform distribution on the hypercube {1,-1}^n), and spherical 2-designs (which are a second-order approximation to the Gaussian distribution). These measurements are easier to implement in experiments, and still ensure successful recovery of most signals, with a few pathological exceptions. (This is joint work with Felix Krahmer at the Technical University of Munich, and Shelby Kimmel at the University of Maryland.)

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