Reduction of logarithmic mask-complexity exponents

Reduce the logarithmic exponents in the mask-complexity requirements for the minimum-norm Gauss–Newton method, namely the $O(\log^2 n)$ masks per independent block required under resampling and the $O(\log^6 n)$ total masks required with fixed data.

Background

The resampled analysis requires O(log⁡2n)O(\log^2 n) masks per independent block, while the fixed-data analysis requires O(log⁡6n)O(\log^6 n) masks in total. The paper explicitly identifies improving these logarithmic sampling exponents as an unresolved direction.

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.

— 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”