Generative modeling via tensor train sketching

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 is constant with respect to the dimension. Finally, we illustrate the performance of the method with several numerical experiments.