On Lipschitz inversion of nonlinear redundant representations

Source abstract

In this note we show 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 : H → R m \alpha : H \rightarrow \mathbb {R}^m is injective, with ( α ( x ) ) k = | ⟨ x , f k ⟩ | 2 (\alpha (x))_k = \lvert \langle x, f_k \rangle \rvert ^2 , where { f 1 , ⋯ , f m } \{f_1, \cdots , f_m\} is a frame for the Hilbert space H H , then there exists a left inverse map ω : R m → H \omega : \mathbb {R}^m \rightarrow H that is Lipschitz continuous. Additionally we obtain that the Lipschitz constant of this inverse map is at most 12 divided by the lower Lipschitz constant of α \alpha .