On stability of generalized phase retrieval and generalized affine phase retrieval

Abstract

In this paper, we consider the stability of intensity measurement mappings corresponding to generalized phase retrieval and generalized affine phase retrieval in the real case. First, we show the bi-Lipschitz property on measurements of noiseless signals. After that, the stability property as regards a noisy signal is given by the Cramer-Rao lower bound.

On stability of generalized (affine) phase retrieval in the complex case

“…Necessary and sufficient conditions as well as minimal number of measurements are given in [10]. The bi-Lipschitz property and CRLB of GPR and GAPR for real signals are discussed in [17]. However, the GPR and GAPR problems with complex signals are also encountered frequently in some fields like optics [15], quantum information [9], interferometry [7], which leads us to addressing the complex case in this paper.…”

Lipschitz Analysis of Generalized Phase Retrievable Matrix Frames

“…Recently several advances have been made in understanding natural generalizations of the problem to arbitrary symmetric measurement matrices [25], unifying the problem of phase retrieval with that of fusion frame reconstruction. Lipschitz stability questions for the generalized phase retrieval are analyzed in [28]. The generalized phase retrieval problem in the case r = 1 has proven amenable to efficient implementations of gradient descent [20] and a probabilistic guarantee of global convergence of first order methods like gradient descent has been obtained in [21] for O(n log 3 (n)) frame vectors.…”