Stable Recovery and Benign Overparameterized Landscapes for Phase Retrieval from Coded Diffraction Patterns

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 $\boldsymbol{x}<em>\star \in \mathbb{C}<sup>n$, let $$\boldsymbol{X}</em>\star=\boldsymbol{x}<em>\star\boldsymbol{x}</em>\star<sup>*$$, 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: $$|\boldsymbol{X}-\boldsymbol{X}<em>\star|_F \lesssim \log²⁽²ⁿ⁾ \frac{|\mathcal A(\boldsymbol{X}-\boldsymbol{X}</em>\star)|<em>2}{\sqrt{nL}}, \boldsymbol{X}\succeq\boldsymbol{0},$$ from which we derive two consequences. First, for $$\boldsymbol{y}=\mathcal A(\boldsymbol{X}</em>\star)+\boldsymbol{e}$$, PhaseLift-type convex programs achieve the Gaussian-type stable recovery bound $$|\widehat{\boldsymbol{X}}-\boldsymbol{X}<em>\star|_F \lesssim \frac{\log<sup>2(2n)}{\sqrt{nL}}|\boldsymbol{e}|_2.$$ Second, in the noiseless case, the nonconvex factorized loss has a benign landscape when the factor width satisfies $r = O(\log<sup>5(2n))$: every second-order critical point $\boldsymbol{V}\in\mathbb C^{n\times r}$ satisfies $$\boldsymbol{V}\boldsymbol{V}^{*=\boldsymbol{X}}</em>\star$$. 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.