### GEORGE MASON
UNIVERSITY

DEPARTMENT OF MATHEMATICAL
SCIENCES

COLLOQUIUM

MARCH 4, 2016

**Speaker: **
Yi-Kai Liu, National Institute of Standards and Technology

**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.

**Time:** Friday, March 4, 2016, 3:30-4:20 p.m.

**Place:** Exploratory Hall, room 4106

**Refreshments** will be served at 3:00 p.m.

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491