exponential lifting

This paper addresses the theoretical performance limits of lifted samplers in Markov chain Monte Carlo (MCMC), specifically analyzing the asymptotic variance relative to conventional reversible base algorithms such as Metropolis–Hastings (MH). Method: Building upon Tierney’s (1998) Markov chain convergence theory, the authors develop a unified framework integrating lifted dynamics modeling, abstraction of direction-inducing mechanisms, and spectral analysis of transition operators. Contribution/Results: The work establishes, for the first time, a universal upper bound: for any target distribution, any direction construction, and any reversible base algorithm (e.g., MH or reversible jump MCMC), the asymptotic variance of a lifted sampler is at most twice that of its base counterpart. This bound is independent of state-space ordering, the specific form of direction definition, or implementation details of the base algorithm—and incurs no additional computational overhead. The result provides a rigorous, general guarantee that lifting yields bounded improvement with controllable risk.

This work proposes a unified framework for sampling from arbitrary log-concave distributions by integrating the In-and-Out algorithm with exponential boosting techniques. By refining the upper bound on the Poincaré constant of the boosted distribution, the method achieves near-optimal convergence rates—matching theoretical lower bounds—for two canonical settings: constrained distributions (e.g., Gaussian restricted to a convex body) and well-conditioned densities (e.g., strongly log-concave and smooth). Combining Poincaré inequality analysis with Markov chain Monte Carlo (MCMC) methodology, the approach efficiently samples from a warm-start initial distribution and establishes nearly tight, unified complexity upper bounds for log-concave sampling.

This work addresses sampling from log-concave distributions under convex constraints and from composite target distributions. By employing an epigraph transformation, both problems are unified as approximate uniform sampling in a higher-dimensional space. The proposed algorithm integrates a proximal sampler, cutting-plane methods, and rejection sampling, relying solely on a separation oracle and a subgradient oracle—without requiring projections, reflections, or barrier functions. This approach yields unbiased and practical constrained sampling and leverages a dual epigraph structure to handle composite objectives in a unified framework. For the first time, mixing-time bounds are established for both problems under Rényi and χ² divergences, enabling efficient sampling even when the geometric structure of the constraint set is unknown.

This work addresses efficient sampling from probability distributions on the spherical manifold. We propose the first geodesic slice sampling Markov chain method specifically designed for the sphere, comprising both a practical, parameter-free, dimension-agnostic contraction variant and a theoretically rigorous ideal version. Our approach constitutes the first extension of slice sampling to non-Euclidean manifolds, leveraging spherical geometry and geodesic dynamics to naturally enforce support constraints. Under mild regularity conditions, we establish uniform ergodicity for the contraction variant. Experiments on Bingham distributions and von Mises–Fisher mixture models demonstrate that our method significantly outperforms random-walk Metropolis–Hastings and Hamiltonian Monte Carlo in mixing efficiency. It thus provides a robust, adaptive, and geometrically aware tool for directional data modeling and shape analysis.

This paper investigates the relationship between the convergence properties of ideal and hybrid slice sampling, specifically focusing on the mutual inheritance of convergence rates under the weak Poincaré inequality framework. Method: We develop a unified Dirichlet form comparison methodology, integrating Markov chain spectral theory with models of slice sampling variants—including stepping-out shrinkage and Hit-and-Run-within-Slice—and systematically apply the weak Poincaré inequality to analyze and compare their convergence behaviors. Contribution/Results: Under mild regularity assumptions, we establish that convergence rates of ideal and hybrid slice samplers are mutually derivable, providing a general theoretical guarantee for convergence transferability. Our framework extends to independent Metropolis–Hastings and multiple slice sampling variants, offering novel theoretical foundations for algorithm selection and design in high-dimensional Bayesian inference.

本文提出MarCo算法，通过从边缘分布进行Metropolis-Hastings采样并结合精确条件分布采样，解决了大尺度问题中MCMC算法计算成本高的问题，证明了其在混合时间和收敛性上的优越性能。

This study addresses the absence of a unified theoretical foundation for stochastic gradient descent (SGD) convergence bounds and data shuffling strategies in high-dimensional machine learning. The proposed approach extends power-law spectral theory to stochastic optimization, demonstrating that the spectral exponent governs SGD dynamics. By integrating high-dimensional statistical learning theory with precise modeling of Gaussian data, tight convergence bounds featuring exact constants are derived. This work represents the first effort to unify abstract spectral theory with practical training sampling mechanisms, rigorously establishing the superiority of single shuffle over alternative sampling schemes. Ultimately, this research elucidates the intrinsic mechanism by which data geometry drives optimization speed, bridges the gap between theory and practice, and provides a comprehensive theoretical framework for stochastic training.

研究二值矩阵固定行和列和的均匀采样问题，提出无拒绝蛇算法，通过生长交替路径并翻转环路，实现高效采样。

Standard Metropolis–Hastings (MH) sampling suffers from high computational cost and slow mixing when drawing samples from sequence-level power distributions \(p^\alpha\), due to a mismatch between its uniform proposal strategy and the spatial sparsity of high-entropy critical regions. This work proposes Entropy-Guided Power Sampling (EGPS), which, for the first time, incorporates token-level entropy into the MCMC proposal mechanism. By guiding local resampling toward uncertain segments and skipping deterministic ones, EGPS focuses computational effort where it is most needed. The method requires no training or external verifier, and its computational overhead scales with entropy magnitude rather than sequence length. Built upon the Multiple-Try Metropolis framework, EGPS achieves state-of-the-art or competitive performance on Qwen2.5-Math-7B across MATH500 (75.8%), HumanEval (62.2%), and GPQA (42.4%), offering up to a 12.6× speedup over the MH baseline.

该研究通过分析不同切片查找方案下混合切片采样的平均目标密度评估次数，提出了自动适应性调优方案，解决了切片采样中初始设置依赖问题。