A geometric analysis of algorithms for phase retrieval and other single-index models with even link functions

Mathematical phase retrieval is the problem of solving systems of quadratic equations. In the first half of this talk, I will discuss recent approaches to this problem, focusing on stochastic gradient methods. In the second half, I will discuss how these algorithms can be extended and generalized. The first extension is to make use of sparsity constraints on the signal vector; the second is to address model ambiguity. In this respect, it is helpful to think of the phase retrieval model as a single-index model with a quadratic link function. In real life applications, it is possible to have link functions that are not exactly quadratic. Possibly, these link functions are also unknown to the observer. I will discuss what can be done to handle such situations.