Invertibility and robustness of phaseless reconstruction

This paper is concerned with the question of reconstructing a vector in a finite-dimensional real Hilbert space when only the magnitudes of the coefficients of the vector under a redundant linear map are known. We analyze various Lipschitz bounds of the nonlinear analysis map and we establish theoretical performance bounds of any reconstruction algorithm. The discussion of robustness is with respect to random noise and with respect to deterministic perturbations. We show that robust and uniformly stable reconstruction is not achievable with the minimum redundancy for phaseless reconstruction. Robust reconstruction schemes require additional redundancy than the critical threshold.