Benign landscape at factor width one

Determine whether the factorized least-squares loss for coded diffraction pattern phase retrieval has a benign landscape at factor width r=1, meaning that every second-order critical point corresponds to the ground-truth lifted matrix and every nonglobal critical point has a direction of strictly negative curvature.

Background

The paper establishes a benign landscape for the factorized coded diffraction pattern loss only under polylogarithmic overparameterization, with factor width r of order log5(n). In contrast, analogous results for Gaussian phase-retrieval measurements can hold at factor width one.

It therefore remains unresolved whether the coded diffraction pattern dependence and structure permit the same benign-landscape property without overparameterization. Resolving this question would clarify whether the additional factor width required by the current analysis is intrinsic or merely technical.

References

Several questions remain open. First, are the logarithmic factors intrinsic, or can one obtain a dimension-independent lower isometry constant? Such a constant would sharpen the stable recovery bound and improve the current logarithmic scalings. Second, does the landscape remain benign at r=1, as in the Gaussian case?

— Stable Recovery and Benign Overparameterized Landscapes for Phase Retrieval from Coded Diffraction Patterns  (2609.30825 - Cai et al., 25 Sep 2026) in Section 6, Discussion and Conclusion (Section 6)