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 three results. First, RGD can be reduced to an effective one-dimensional recursion which mirrors the evolution of the AMP iterates. Second, from a warm start, RGD rapidly converges to a fixed point in correlation space, which recovers Bayes-optimal performance when run on the posterior. Third, conditioned on widely believed mixing results for the SK model, we recover the phase transition for non-trivial inference. The full version of this paper can be found on arXiv (arXiv ID: 2512.02384).

Many natural Markov chains fail to mix to their stationary distribution in polynomially many steps. Often, this slow mixing is inevitable since it is computationally intractable to sample from their stationary measure. Nevertheless, Markov chains can be shown to always converge quickly to measures that are locally stationary, i.e., measures that don't change over a small number of steps. These locally stationary measures are analogous to local minima in continuous optimization, while stationary measures correspond to global minima. While locally stationary measures can be statistically far from stationary measures, do they enjoy provable theoretical guarantees that have algorithmic implications? We study this question in this work and demonstrate three algorithmic applications of locally stationary measures: 1)We show that Glauber dynamics on the hardcore model can be used to find large independent sets in triangle-free graphs of bounded degree. 2)We prove that Glauber dynamics on the Ising model defined by a spiked matrix model finds a vector with constant correlation with the planted spike. 3)We show that for sufficiently large constant signal-to-noise ratio, Glauber dynamics on the Ising model finds a vector that has constant correlation with the hidden community vector. In other words, Glauber dynamics subsumes the spectral method for spiked Wigner and community detection, by weakly recovering the planted spike. The full version of this paper can be found on arXiv(arXiv ID: 2405.20849).

Motivated by the community detection problem in Bayesian inference, as well as the recent explosion of interest in spin glasses from statistical physics, we study the classical Glauber dynamics for sampling from Ising models with sparse random interactions. It is now well-known that when the in- teraction matrix has spectral diameter less than 1, Glauber dynamics mixes in near-linear time. Unfortunately, such criteria fail dramatically for interactions supported on arguably the most well-studied sparse random graph: the Erdos-Renyi random graph. There is a scarcity of positive results in this setting due to the presence of almost linearly many outlier eigenvalues of unbounded magnitude. We prove that for the Viana-Bray spin glass, where the interactions are supported on a random graph and randomly assigned signs, Glauber dynamics mixes in almost-linear time with high probability at sufficiently high temperatures, and we conjecture that our results are tight up to constants. We further extend our results to random graphs drawn according to the 2-community stochastic block model, as well as when the interactions are given by a “centered” version of the adjacency matrix. The latter setting is particularly relevant for the inference problem in community detection. Indeed, we build on this result to demonstrate that Glauber dynamics succeeds at recovering communities in the stochastic block model in a companion paper. The primary technical ingredient in our proof is showing that with high probability, a sparse random graph can be decomposed into two parts - a bulk which behaves like a graph with bounded maximum degree and a well-behaved spectrum, and a near- forest with favorable pseudorandom properties. We then use this decomposition to design a localization procedure that interpolates to simpler Ising models supported only on the near-forest, and then execute a pathwise analysis to establish a modified log- Sobolev inequality. The full version of this paper can be found on arXiv (arXiv ID: 2405.06616).

We study the problem of robust community recovery: efficiently recovering communities in sparse stochastic block models in the presence of adversarial corruptions. In the absence of adversarial corruptions, there are efficient algorithms when the signal-to-noise ratio exceeds the Kesten–Stigum (KS) threshold, widely believed to be the computational threshold for this problem. The question we study is: does the computational threshold for robust community recovery also lie at the KS threshold? We answer this question affirmatively, providing an algorithm for robust community recovery for arbitrary stochastic block models on any constant number of communities, generalizing the work of Ding, d’Orsi, Nasser & Steurer on an efficient algorithm above the KS threshold in the case of 2-community block models. There are three main ingredients to our work: (1) The Bethe Hessian of the graph is defined as HG(t) ≜ (DG−I)t2 − AGt + I where DG is the diagonal matrix of degrees and AG is the adjacency matrix. Empirical work suggested that the Bethe Hessian for the stochastic block model has outlier eigenvectors corresponding to the communities right above the Kesten-Stigum threshold. We formally confirm the existence of outlier eigenvalues for the Bethe Hessian, by explicitly constructing outlier eigenvectors from the community vectors. (2) We develop an algorithm for a variant of robust PCA on sparse matrices. Specifically, an algorithm to partially recover top eigenspaces from adversarially corrupted sparse matrices under mild delocalization constraints. (3) A rounding algorithm to turn vector assignments of vertices into a community assignment, inspired by the algorithm of Charikar & Wirth for 2XOR.