Analysis of Learning a Flow-based Generative Model from Limited Sample Complexity

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 means of the generated mixture and the mean of the target mixture, which we show decays as Θn( 1 /n). Finally, this rate is shown to be in fact Bayes-optimal. • We provide a sharp asymptotic closed-form characterization of the learnt velocity field, as a function of the target Gaussian mixture parameters, the stochastic interpolation schedule, and the number of training samples n. • We characterize the associated flow by providing a tight characterization of a small number of summary statistics, tracking the dynamics of a sample from the Gaussian base distribution as it is transported by the learnt velocity