FRAMES AND PHASELESS RECONSTRUCTION AMS SHORT COURSE: JOINT MATHEMATICS MEETINGS . . .

Frame design for phaseless reconstruction is now part of the broader problem of nonlinear recon- struction and is an emerging topic in harmonic analysis. The problem of phaseless reconstruction can be simply stated as follows. Given the magnitudes of the coefficients of an output of a linear redundant system (frame), we want to reconstruct the unknown input. This problem has first occurred in X-ray crystallography starting from the early 20th century. The same nonlinear reconstruction problem shows up in speech processing, particularly in speech recognition. In this lecture we shall cover existing analysis results as well as algorithms for signal recovery including: necessary and sufficient conditions for injectivity, Lipschitz bounds of the nonlinear map and its left inverses, stochastic performance bounds, convex relaxation algorithms for inversion, least-squares inversion algorithms