IAM-PIMS Distinguished Colloquium: Thomas Strohmer

Phase retrieval is the century-old problem of reconstructing a function, such as a signal or image, from intensity measurements, typically from the modulus of a diffracted wave. Phase retrieval problems - which arise in numerous areas including X-ray crystallography, astronomy, diffraction imaging and quantum physics - are notoriously difficult to solve numerically. They also pervade many areas of mathematics, such as numerical analysis, harmonic analysis, algebraic geometry, combinatorics, and differential geometry. I will introduce a novel framework for phase retrieval, which comprises tools from optimization, random matrix theory, and compressive sensing. In particular, we will see that for certain types of random measurements a function, such as a signal or image, can be recovered exactly with high probability by solving a convenient semidefinite program without any assumption about the function whatsoever and under a mild condition on the number of measurements. Our method, known as PhaseLift, is also provably stable vis-a-vis noise. I will describe how this approach carries over to the classical phase retrieval setting using structured random illuminations. I conclude with some open problems.