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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.07816 |
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| _version_ | 1866917133118603264 |
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| author | Kogan, David |
| author_facet | Kogan, David |
| contents | We prove universality for Approximate Message Passing (AMP) with polynomial nonlinearities applied to symmetric sub-Gaussian matrices $A\in\mathbb R^{N\times N}$. Our approach is combinatorial: we represent AMP iterates as sums over trees and define a Wick pairing algebra that counts the number of valid row-wise pairings of edges. The number of such pairings coincides with the trees contribution to the state evolution formulas. This algebra works for non-Gaussian entries. For polynomial nonlinearities of degree at most $D$, we show that the moments of AMP iterates match their state evolution predictions for $t \lesssim \frac{\log N}{D\log D}$ iterations. The proof controls all "excess" trees via explicit enumeration bounds, showing non "Wick-paired" contributions vanish in the large-$N$ limit. The same framework should apply, with some modifications, to spiked AMP and tensor AMP models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07816 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universality of AMP via Tree Pairings Kogan, David Probability We prove universality for Approximate Message Passing (AMP) with polynomial nonlinearities applied to symmetric sub-Gaussian matrices $A\in\mathbb R^{N\times N}$. Our approach is combinatorial: we represent AMP iterates as sums over trees and define a Wick pairing algebra that counts the number of valid row-wise pairings of edges. The number of such pairings coincides with the trees contribution to the state evolution formulas. This algebra works for non-Gaussian entries. For polynomial nonlinearities of degree at most $D$, we show that the moments of AMP iterates match their state evolution predictions for $t \lesssim \frac{\log N}{D\log D}$ iterations. The proof controls all "excess" trees via explicit enumeration bounds, showing non "Wick-paired" contributions vanish in the large-$N$ limit. The same framework should apply, with some modifications, to spiked AMP and tensor AMP models. |
| title | Universality of AMP via Tree Pairings |
| topic | Probability |
| url | https://arxiv.org/abs/2512.07816 |