In this paper, we introduce a sketching algorithm for constructing a tensor train representation of a probability density from its samples. Our method deviates from the standard recursive SVD-based procedure for constructing a tensor train. Instead, we formulate and solve a sequence of small linear systems for the individual tensor train cores. This approach can avoid the curse of dimensionality that threatens both the algorithmic and sample complexities of the recovery problem. Specifically, for Markov models, we prove that the tensor cores can be recovered with a sample complexity that scales logarithmically in the dimensionality. Finally, we illustrate the performance of the method with several numerical experiments.

翻译：本文提出一种草绘算法，用于从概率密度的样本构建其张量列车表示。该方法不同于传统的基于递归奇异值分解的张量列车构建流程，而是通过构建并求解一系列小型线性系统来获取各个张量列车核。这种策略能够避免维度灾难——该问题同时威胁着算法复杂度与样本复杂度。具体而言，对于马尔可夫模型，我们证明张量核的恢复样本复杂度随维度呈对数增长。最后，通过多项数值实验验证了该方法的性能。