Generative modeling via tensor train sketching
This work addresses the curse of dimensionality in generative modeling for high-dimensional data, offering a more efficient method for density estimation, though it appears incremental as it builds on existing tensor train frameworks.
The paper tackles the problem of constructing tensor train representations of probability densities from samples by introducing a sketching algorithm that solves small linear systems for tensor cores, avoiding the curse of dimensionality. It proves logarithmic sample complexity scaling in dimensionality for Markov models and demonstrates performance through numerical experiments.
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.