Accurate stable recovery for phase retrieval using coded diffraction patterns
Stable Recovery and Benign Overparameterized Landscapes for Phase Retrieval from Coded Diffraction Patterns
Information Theory
Summary
The paper deals with the problem of reconstructing a signal from intensity measurements, specifically using coded diffraction patterns. The authors show that with a small number of random masks, their method can reliably and stably recover the original signal even when there is noise. They also demonstrate that certain mathematical formulations used to find the signal have no troublesome false solutions under these conditions. This helps guarantee that recovery algorithms work well in practice for these types of problems.
What this means in practice
- •For computational imaging engineers: Design more reliable imaging systems that reconstruct phase information from coded diffraction measurements with theoretical stability guarantees.
- •For signal processing software developers: Implement algorithms for phase retrieval that avoid false solutions and maintain accuracy under moderate noise using guaranteed sample complexity.
A theory result. No direct application yet.
Authors
Jian-Feng Cai, Zhibo Jin, Tong Wu, Ruizhe Xia
Abstract
Coded diffraction patterns (CDPs) provide a structured and physically relevant model for phase retrieval, but the dependence among Fourier measurements generated by a common mask makes sharp stability analysis challenging. For a fixed unit-norm signal $\boldsymbol{x}_\star \in \mathbb{C}^n$, let $\boldsymbol{X}_\star=\boldsymbol{x}_\star\boldsymbol{x}_\star^*$, and let $\mathcal A$ be the lifted CDP measurement operator. We prove that, with $L=O(\log n)$ random masks, the following uniform lower isometry holds with high probability: $\|\boldsymbol{X}-\boldsymbol{X}_\star\|_F \lesssim \log^2(2n) \frac{\|\mathcal A(\boldsymbol{X}-\boldsymbol{X}_\star)\|_2}{\sqrt{nL}}, \boldsymbol{X}\succeq\boldsymbol{0},$ from which we derive two consequences. First, for $\boldsymbol{y}=\mathcal A(\boldsymbol{X}_\star)+\boldsymbol{e}$, PhaseLift-type convex programs achieve the Gaussian-type stable recovery bound $\|\widehat{\boldsymbol{X}}-\boldsymbol{X}_\star\|_F \lesssim \frac{\log^2(2n)}{\sqrt{nL}}\|\boldsymbol{e}\|_2.$ Second, in the noiseless case, the nonconvex factorized loss has a benign landscape when the factor width satisfies $r = O(\log^5(2n))$: every second-order critical point $\boldsymbol{V}\in\mathbb C^{n\times r}$ satisfies $\boldsymbol{V}\boldsymbol{V}^*=\boldsymbol{X}_\star$. The key ingredient is a uniform operator-norm bound over row subsets of the dependent CDP measurement matrix, which permits the removal of a controlled set of adaptively selected rows while preserving tangent injectivity.