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Capture AMP with degree-O(log n) polynomials

Develop a rigorous method showing that approximate message passing (AMP) algorithms requiring O(log n) iterations or a spectral initialization can be implemented by degree-O(log n) polynomials in the input, for example by performing O(log n) rounds of power iteration followed by O(1) AMP iterations, thereby establishing matching low-degree upper bounds for these AMP settings.

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Background

AMP provides state-of-the-art iterative algorithms for many planted estimation and non-planted optimization problems, often matching sharp statistical predictions. Existing analyses show that O(1) AMP iterations can be captured by constant-degree polynomials, but some AMP variants need O(log n) iterations or spectral initialization.

Formalizing a degree-O(log n) polynomial representation of such AMP variants would bolster the degree–runtime correspondence and supply low-degree upper bounds that match AMP performance in these regimes.

References

It is a good open question to prove that degree-$O(\log n)$ polynomials capture AMP in these settings, perhaps using $O(\log n)$ rounds of power iteration followed by $O(1)$ iterations of AMP.

— Computational Complexity of Statistics: New Insights from Low-Degree Polynomials (2506.10748 - Wein, 12 Jun 2025) in Section 6.2 (Algorithms Captured by Polynomials), item “Approximate Message Passing (AMP)”