Title: Phase Retrieval meets Statistical Learning Theory

Speaker: Justin K. Romberg (School of Electrical and Computer Engineering, GA Tech)

Date/Time: Wednesday 2/15/2017, 12n – 1pm, (Lunch at 11:30am)

Location: TSRB Banquet Hall (85 5th Street NE, Atlanta, GA 30309)

Abstract:

We will present a new convex relaxation for the classic problem of phase retrieval, where we want to reconstruct an unknown signal from observations of the *magnitude* of a series of linear measurements. The problem is nonlinear, amounts to solving a system of quadratic equations. Our method for solving this problem is based on linear (in the real case) or quadratic (in the complex case) programming. The number of variables in the convex relaxation is the same as the dimension of the signal, so we avoid the increase in dimensionality inherent to lifting schemes based on semidefinite programming. Our method has a clear geometric interpretation: by relaxing each magnitude measurement into a pair of inequality constraints, we know that the signal of interest lies on the surface of the polytope defined by the intersection of these constraints. We show that even a rough guess of where the signal lies (a guess that can itself be formed from the measured data) is enough to define a linear functional over this polytope whose maximum is at the true signal. Our analysis uses classical results from statistical learning theory, in particular, the VC dimension and generalization bound.

We will also discuss very recent work on how these techniques can be extended to provide convex relaxations for solving more general systems of nonlinear equations.

This is joint work with Sohail Bahmani.

Bio:

Dr. Justin Romberg is a professor in the School of Electrical and Computer Engineering at the Georgia Institute of Technology. Dr. Romberg received the B.S.E.E. (1997), M.S. (1999) and Ph.D. (2004) degrees from Rice University in Houston, Texas. From fall 2003 until fall 2006, he was a Postdoctoral Scholar in Applied and Computational Mathematics at the California Institute of Technology. He spent the summer of 2000 as a researcher at Xerox PARC, the fall of 2003 as a visitor at the Laboratoire Jacques-Louis Lions in Paris, and the fall of 2004 as a Fellow at UCLA’s Institute for Pure and Applied Mathematics. In fall 2006, he joined the ECE faculty as a member of the Center for Signal and Image Processing. In 2008, he received an ONR Young Investigator Award, in 2009 he received a PECASE award and a Packard Fellowship, and in 2010 he was named a Rice University Outstanding Young Engineering Alumnus. In 2006-2007, he was a consultant for the TV show “Numb3rs,” and from 2008-2011, he was an Associate Editor for the *IEEE Transactions on Information Theory*.