The possibility of using the Eulerian discretization for the problem of modelling high-dimensional distributions and sampling, is studied. The problem is posed as a minimization problem over the space of probability measures with respect to the Wasserstein distance and solved with entropy-regularized JKO scheme. Each proximal step can be formulated as a fixed-point equation and solved with accelerated methods, such as Anderson's. The usage of low-rank Tensor Train format allows to overcome the \emph{curse of dimensionality}, i.e. the exponential growth of degrees of freedom with dimension, inherent to Eulerian approaches. The resulting method requires only pointwise computations of the unnormalized posterior and is, in particular, gradient-free. Fixed Eulerian grid allows to employ a caching strategy, significally reducing the expensive evaluations of the posterior. When the Eulerian model of the target distribution is fitted, the passage back to the Lagrangian perspective can also be made, allowing to approximately sample from it. We test our method both for synthetic target distributions and particular Bayesian inverse problems and report comparable or better performance than the baseline Metropolis-Hastings MCMC with same amount of resources. Finally, the fitted model can be modified to facilitate the solution of certain associated problems, which we demonstrate by fitting an importance distribution for a particular quantity of interest.

翻译：本文研究了采用欧拉离散化方法处理高维分布建模与采样问题的可能性。该问题被表述为概率测度空间上关于Wasserstein距离的最小化问题，并通过熵正则化的JKO格式求解。每个邻近步可表述为不动点方程，并可采用加速方法（如Anderson法）求解。低秩张量列格式的使用能够克服欧拉方法固有的维度灾难问题，即自由度随维度呈指数增长的特性。所得方法仅需对未归一化后验分布进行逐点计算，尤其无需梯度信息。固定的欧拉网格支持缓存策略，显著减少了昂贵的后验分布计算次数。当目标分布的欧拉模型拟合完成后，可转换回拉格朗日视角，从而实现近似采样。我们通过合成目标分布和特定贝叶斯反问题测试本方法，结果表明在相同资源消耗下，其性能与基准Metropolis-Hastings MCMC方法相当或更优。最后，可通过修改拟合模型以辅助求解特定关联问题，本文通过针对特定感兴趣量拟合重要性分布验证了该特性。