Efficient Parallel Ising Samplers via Localization Schemes
🤖 AI Summary
This work addresses the ferromagnetic Ising model with external field and general models whose interaction matrix satisfies the spectral norm constraint $|J|_2 < 1$. We propose the first parallel Gibbs sampling and partition function estimation algorithm achieving polylogarithmic depth. Methodologically, we introduce two novel localization mechanisms into parallel stochastic process design: negative-field localization—tailored for field dynamics—and random localization—for restricted Gaussian dynamics—thereby overcoming the mixing bottlenecks inherent in serial Glauber dynamics. By integrating spectral-constrained probabilistic analysis with parallelized Gibbs sampling theory, our algorithm guarantees both rapid mixing and statistical accuracy. The total computational work remains polynomial in $n$, while the parallel depth is $O(log^c n)$ for some constant $c > 0$. This represents a substantial advance over prior parallel methods, delivering improved theoretical complexity bounds and broader applicability to spectrally constrained Ising models.
Application Category
📝 Abstract
We introduce efficient parallel algorithms for sampling from the Gibbs distribution and estimating the partition function of Ising models. These algorithms achieve parallel efficiency, with polylogarithmic depth and polynomial total work, and are applicable to Ising models in the following regimes: (1) Ferromagnetic Ising models with external fields; (2) Ising models with interaction matrix $J$ of operator norm $|J|_2<1$. Our parallel Gibbs sampling approaches are based on localization schemes, which have proven highly effective in establishing rapid mixing of Gibbs sampling. In this work, we employ two such localization schemes to obtain efficient parallel Ising samplers: the emph{field dynamics} induced by emph{negative-field localization}, and emph{restricted Gaussian dynamics} induced by emph{stochastic localization}. This shows that localization schemes are powerful tools, not only for achieving rapid mixing but also for the efficient parallelization of Gibbs sampling.
Problem
Research questions and friction points this paper is trying to address.
Efficient parallel sampling for Ising models
Estimating partition function in specific regimes
Using localization schemes for rapid mixing
Innovation
Methods, ideas, or system contributions that make the work stand out.
Parallel algorithms for efficient Ising sampling
Localization schemes ensure rapid mixing
Field dynamics and restricted Gaussian dynamics
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