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 under natural conditions, 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.

在本文中，我们介绍了一种草图算法，用于从样本构建概率密度的张量序列表示。我们的方法偏离了构建张量序列的标准递归 SVD 程序。相反，我们为各个张量序列核心制定并求解一系列小型线性系统。这种方法可以避免维数灾难，该灾难威胁到恢复问题的算法和样本复杂性。具体来说，对于自然条件下的马尔可夫模型，我们证明可以通过在维度上以对数方式缩放的样本复杂度来恢复张量核心。最后，我们通过几个数值实验说明了该方法的性能。