Analysis of learning a flow-based generative model from limited sample
  complexity
Hugo Cui Florent Krzakala Eric Vanden-Eijnden Lenka Zdeborová
We study the problem of training a flow-based generative model, parametrized
by a two-layer autoencoder, to sample from a high-dimensional Gaussian mixture.
We provide a sharp end-to-end analysis of the problem. First, we provide a
tight closed-form characterization of the learnt velocity field, when
parametrized by a shallow denoising auto-encoder trained on a finite number $n$
of samples from the target distribution. Building on this analysis, we provide
a sharp description of the corresponding generative flow, which pushes the base
Gaussian density forward to an approximation of the target density. In
particular, we provide closed-form formulae for the distance between the mean
of the generated mixture and the mean of the target mixture, which we show
decays as $\Theta_n(\frac{1}{n})$. Finally, this rate is shown to be in fact
Bayes-optimal.