Generative modeling via tensor train sketching

TL;DR

This paper presents a novel tensor train sketching algorithm for efficiently constructing tensor train representations of probability densities from samples, avoiding the curse of dimensionality and demonstrating favorable sample complexity.

Contribution

The paper introduces a new linear systems-based tensor train construction method that reduces sample complexity and computational challenges compared to traditional SVD-based approaches.

Findings

Tensor cores can be recovered with logarithmic sample complexity in dimension.

The method outperforms standard recursive SVD procedures.

Numerical experiments validate the effectiveness of the approach.

Abstract

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.