We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyv\"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.

翻译：我们提出了一种基于各向同性高斯核（固定单一噪声尺度）平滑方案的理论框架，用于从非归一化密度中进行采样。我们证明，通过累积等噪声水平的噪声测量，可将从某密度（对该密度仅需极弱假设）的采样分解为一系列从对数凹条件密度进行的采样序列。该构造的独特性在于记录了样本历史轨迹，使整体过程具有非马尔可夫性，但算法层面仍保持轻量——历史仅通过样本运行经验均值的形式体现。我们的采样算法推广了“跳步采样”(Saremi & Hyvärinen, 2019)：其中“行走”阶段演变为（对数凹）马尔可夫链构成的（非马尔可夫）链，而基于累积测量值的“跳跃”通过经验贝叶斯方法实现。我们采用2- Wasserstein度量对采样算法进行定量分析，并与多种朗之万MCMC算法进行对比，同时报道了该算法在分布模态间实现“隧穿”的卓越能力。