Phase Retrieval using Lipschitz Continuous Maps
This provides a theoretical guarantee for stable reconstruction in signal processing, though it is incremental as it builds on known injectivity conditions.
The paper tackles the phase retrieval problem by proving that reconstruction from magnitudes of frame coefficients can be achieved using Lipschitz continuous maps, showing the existence of a left inverse map with a Lipschitz constant independent of space dimension or frame redundancy.
In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map $α:{\mathcal H}\rightarrow\mathbb{R}^m$ is injective, with $(α(x))_k=|<x,f_k>|^2$, where $\{f_1,\ldots,f_m\}$ is a frame for the Hilbert space ${\mathcal H}$, then there exists a left inverse map $ω:\mathbb{R}^m\rightarrow {\mathcal H}$ that is Lipschitz continuous. Additionally we obtain the Lipschitz constant of this inverse map in terms of the lower Lipschitz constant of $α$. Surprisingly the increase in Lipschitz constant is independent of the space dimension or frame redundancy.