We present a tensor train (TT) based algorithm designed for sampling from a target distribution and employ TT approximation to capture the high-dimensional probability density evolution of overdamped Langevin dynamics. This involves utilizing the regularized Wasserstein proximal operator, which exhibits a simple kernel integration formulation, i.e., the softmax formula of the traditional proximal operator. The integration, performed in $\mathbb{R}^d$, poses a challenge in practical scenarios, making the algorithm practically implementable only with the aid of TT approximation. In the specific context of Gaussian distributions, we rigorously establish the unbiasedness and linear convergence of our sampling algorithm towards the target distribution. To assess the effectiveness of our proposed methods, we apply them to various scenarios, including Gaussian families, Gaussian mixtures, bimodal distributions, and Bayesian inverse problems in numerical examples. The sampling algorithm exhibits superior accuracy and faster convergence when compared to classical Langevin dynamics-type sampling algorithms.

翻译：我们提出了一种基于张量列车（TT）的算法，用于从目标分布中采样，并利用TT近似捕捉过阻尼朗之万动力学的高维概率密度演化。该方法涉及正则化Wasserstein近端算子的应用，该算子具有简单的核积分形式，即传统近端算子的softmax公式。由于积分在$\mathbb{R}^d$空间中进行，实际场景中面临挑战，使得该算法仅在TT近似的辅助下才具有实际可行性。在高斯分布的具体情境下，我们严格证明了采样算法对目标分布的无偏性与线性收敛性。为评估所提方法的有效性，我们将其应用于数值案例中的多种场景，包括高斯族分布、高斯混合分布、双峰分布以及贝叶斯反问题。与经典朗之万动力学类采样算法相比，该采样算法展现出更高的精度和更快的收敛速度。