论文标题
通过张量火车素描的生成建模
Generative modeling via tensor train sketching
论文作者
论文摘要
在本文中，我们介绍了一种草图算法，用于从其样品中构建概率密度的张量列表表示。我们的方法偏离了基于标准的递归SVD构建张量火车的程序。取而代之的是，我们为单个张量火车芯制定并解决了一系列小型线性系统。这种方法可以避免维数的诅咒，从而威胁恢复问题的算法和样本复杂性。具体而言，对于在自然条件下的马尔可夫模型，我们证明可以以样品复杂性在维度上进行对数缩放。最后，我们通过几个数值实验说明了该方法的性能。
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.
