提出一种新型粒子采样方法，提升收敛速度与稳定性。
Preconditioned Regularized Wasserstein Proximal Sampling
- 用正则化Wasserstein近点算子逼近得分函数，结合预条件加速
- 对二次势能系统给出非渐近收敛分析，偏差仅依赖正则化参数
- 适合高维复杂分布采样，如图像去模糊和贝叶斯神经网络训练
本文研究通过有限粒子演化从吉布斯分布中采样。提出一种近期无噪声采样方法的预条件版本，该方法通过数值可处理的正则化Wasserstein近点算子得分函数近似实现。该方法由耦合各向异性热方程的Cole--Hopf变换导出，得到预条件正则化Wasserstein近点的核形式。所提方法的扩散项可解释为类似Transformer架构的改进自注意力模块。对于二次势能，给出了离散时间非渐近收敛分析，并显式刻画了偏差——其依赖于正则化参数但与步长无关。实验表明，在多种对数凹与非对数凹样本及贝叶斯总变差正则化图像去模糊任务中，该方法表现出加速效果与粒子级稳定性；在使用可变预条件矩阵时，于非凸贝叶斯神经网络训练中表现竞争或更优。
原文摘要 · Abstract (English)
We consider sampling from a Gibbs distribution by evolving finitely many particles. We propose a preconditioned version of a recently proposed noise-free sampling method, governed by approximating the score function with the numerically tractable score of a regularized Wasserstein proximal operator. This is derived by a Cole--Hopf transformation on coupled anisotropic heat equations, yielding a kernel formulation for the preconditioned regularized Wasserstein proximal. The diffusion component of the proposed method is also interpreted as a modified self-attention block, as in transformer architectures. For quadratic potentials, we provide a discrete-time non-asymptotic convergence analysis and explicitly characterize the bias, which is dependent on regularization and independent of step-size. Experiments demonstrate acceleration and particle-level stability on various log-concave and non-log-concave toy examples to Bayesian total-variation regularized image deconvolution, and competitive/better performance on non-convex Bayesian neural network training when utilizing variable preconditioning matrices.
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