Convergence from random initialization

Establish convergence of the minimum-norm Gauss–Newton method for coded diffraction phase retrieval when initialized randomly rather than by spectral initialization.

Background

All convergence guarantees in the paper rely on a spectral initializer that places the first iterate inside a suitable local basin. The authors leave unresolved whether convergence can be proved when the algorithm starts from a random initialization.

References

Several questions remain open. First, our guarantees hold for each fixed signal $\star$, rather than simultaneously for all $\star$. Extending the analysis to a uniform recovery guarantee would be of interest. Second, the mask complexities are $O(\log2 n)$ per independent block with resampling and $O(\log6 n)$ in total with fixed data. Reducing these logarithmic exponents is another important direction. Third, our convergence guarantees rely on spectral initialization; establishing convergence from a random initialization remains an interesting challenge.

— The Gauss--Newton Method for Phase Retrieval from Coded Diffraction Patterns  (2610.18021 - Huang et al., 13 Oct 2026) in Section 6, “Conclusions and discussion”