The phaseless reconstruction problem can be stated as follows:

Given the magnitudes of a vector coefficients with respect to a linear redundant system (frame), we want to reconstruct the unknown vector. 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 talk we present a Lipschitz analysis of the problem as well as Cramer-Rao Lower Bounds that govern any reconstruction algorithm. In particular we show that the left inverse of the nonlinear analysis map can be extended to the entire measurement space with a small increase in the Lipschitz constant independent of the dimension of the space or the frame redundancy. This is joint work with Dongmian Zou.