approximate sampling algorithm

This work addresses local sampling from high-dimensional distributions—such as Ising models—overcoming the fundamental limitation of existing algorithms that require a “local uniformity” assumption. We propose the first local Gibbs sampler that operates without this assumption, built upon a local query framework, local induced distribution analysis for high-dimensional models, and adaptive Markov chain construction. Our method enables efficient approximate sampling from near-critical spin systems on unbounded-degree graphs. Theoretically, it achieves exponential speedup in local time complexity compared to prior state-of-the-art methods, and natively supports dynamic graph updates. Crucially, this is the first algorithm to achieve truly local, efficient sampling in the near-critical regime—where correlations become long-range—thereby significantly extending the applicability boundary of local sampling techniques to previously intractable parameter regimes.

This work addresses the limitation of sequential adaptive sampling in approximate counting within parallel computation by introducing efficient non-adaptive and two-round adaptive sampling strategies. The authors reformulate the counting problem as estimating ratios of partition functions over a family of Gibbs distributions. Leveraging simulated annealing within the randomized NC (RNC) parallel framework, they present the first non-adaptive algorithm with sample complexity $O(q \log^2 h / \varepsilon^2)$ and design a two-round adaptive scheme achieving $O(q \log h / \varepsilon^2)$, approaching the theoretical optimum while significantly enhancing parallelizability. The proposed methods are successfully applied to achieve efficient approximate counting for antiferromagnetic two-spin systems, monomer-dimer models, and ferromagnetic Ising models.

This paper presents the first efficient distributed approximation algorithm for computing the partition function of graphical Gibbs distributions in the Congested Clique model, unifying counting problems for combinatorial structures such as $q$-colorings and independent sets. Methodologically, it introduces, for the first time in this model, the theoretical connection between combinatorial sampling and counting, establishing a generic framework grounded in locality and rapid mixing conditions; it further enables parallel Markov chain sampling via triangle counting and semiring matrix multiplication. Theoretical contributions include: (i) an $widetilde{O}(n^{1/3}/varepsilon^2)$-round algorithm for $q$-coloring counting when $q > 2Delta$; and (ii) convergence to $varepsilon$-accuracy for the hard-core model within $widetilde{O}(1/varepsilon^2)$ rounds under fugacity $lambda leq alpha/(Delta-1)$ with $alpha < 1$, substantially improving upon prior distributed results.

This paper addresses the problem of efficiently approximating the partition function of Gibbs distributions in parallel. Existing algorithms face a fundamental trade-off between total work and parallel depth. We propose the first counting-to-sampling reduction framework that simultaneously achieves near-optimal total work—approaching the information-theoretic lower bound—and logarithmic parallel depth. Methodologically, we build upon the simulated annealing paradigm, designing an adaptive temperature schedule and a parallel sampling verification mechanism to ensure work efficiency. Our theoretical contribution is the first work-efficient parallel counting algorithm for both the hardcore model and the Ising model within their uniqueness regimes. This overcomes the inherent limitations of prior approaches: non-adaptive algorithms suffer from suboptimal work complexity, while adaptive ones are inherently sequential. Our framework thus unifies asymptotic optimality in work with high parallelism, establishing a new state of the art in parallel approximate counting.

This study addresses the lack of efficient sampling and partition function approximation algorithms for ferromagnetic two-state systems within certain parameter regimes. By constructing weighted subgraph and random-cluster-like models, the authors establish novel equivalences between these combinatorial structures and the target physical system. Leveraging these connections, they propose the first efficient sampling algorithm tailored to this regime and design a partition function approximation algorithm running in nearly quadratic time on bounded-degree graphs and in polynomial time on general graphs. This represents a significant improvement over the method of Guo et al. (2020). The work integrates techniques from graph theory, statistical physics, and randomized approximation to deliver an enhanced computational framework for analyzing such systems.

This work addresses the longstanding challenge in blue-noise sampling of simultaneously achieving high quality, locality, and parallelizability. The authors formulate blue-noise generation as a lattice-based Gibbs distribution with pairwise repulsive interactions and, for the first time, unify it within a parametric Gibbs ensemble framework that enables continuous control over spectral characteristics. By introducing bounded dependency regions and a haloed tiling strategy combined with the Coupling From The Past algorithm, they achieve communication-free exact sampling per tile, with memory consumption scaling only with tile size. Experiments demonstrate that the method faithfully reproduces standard blue-noise spectra, yields tilewise results bit-identical to global generation, and successfully scales to applications such as 14K adaptive stippling and multi-class extensions.

This work addresses the challenges of sampling from high-dimensional probability distributions, which are often hindered by the curse of dimensionality and metastable multimodal traps. It introduces a novel approach that repurposes generative models—such as normalizing flows and diffusion models—from their conventional data-driven paradigm into data-free auxiliary tools for efficient and accurate sampling from target distributions known only up to an unnormalized density. By integrating Monte Carlo methods with enhanced sampling techniques, the authors develop a tailored training strategy and systematically formulate a unified framework for generative-model-assisted sampling. This contribution offers a theoretically grounded and practically implementable tutorial, serving as both a methodological guide and a springboard for interdisciplinary research at the intersection of physics and machine learning.

This work investigates the slow mixing of Markov chains for the Potts model on sparse random graphs $G(n,d/n)$ at low temperatures, where the ordered phase dominates. Focusing on the local limit—a Poisson Galton–Watson tree with monochromatic boundary conditions—the authors extend the rapid mixing results previously established for regular trees to the irregular-degree setting of Poisson trees. By developing a refined analytical framework based on adaptive block decomposition, decay of correlations estimates, and functional inequalities, they establish near-linear mixing time for Glauber dynamics in the low-temperature regime. This analysis yields the first near-linear-time approximate sampling algorithm for the Potts model on $G(n,d/n)$ that is valid across all temperatures.