Abstract:Markov chain Monte Carlo algorithms have long been observed to obtain near-optimal performance in various Bayesian inference settings. However, developing a supporting theory that makes these studies rigorous has proved challenging.
In this paper, we study the classical spiked Wigner inference problem, where one aims to recover a planted Boolean spike from a noisy matrix measurement. We relate the recovery performance of Glauber dynamics on the annealed posterior to the performance of Approximate Message Passing (AMP), which is known to achieve Bayes-optimal performance. Our main results rely on the analysis of an auxiliary Markov chain called restricted Gaussian dynamics (RGD). Concretely, we establish the following results:
1. RGD can be reduced to an effective one-dimensional recursion which mirrors the evolution of the AMP iterates.
2. From a warm start, RGD rapidly converges to a fixed point in correlation space, which recovers Bayes-optimal performance when run on the posterior.
3. Conditioned on widely believed mixing results for the SK model, we recover the phase transition for non-trivial inference.
From: Amit Rajaraman [view email]
[v1]
Tue, 2 Dec 2025 03:54:01 UTC (71 KB)
[v2]
Wed, 3 Dec 2025 04:13:55 UTC (69 KB)
[v3]
Fri, 17 Jul 2026 16:38:29 UTC (71 KB)
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