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.

翻译：我们研究了训练基于流的生成模型的问题，该模型由两层自编码器参数化，旨在从高维高斯混合分布中采样。我们对该问题提供了精确的端到端分析。首先，当速度场由基于目标分布中有限数量$n$个样本训练的浅层去噪自编码器参数化时，我们给出了所学速度场的紧致闭式特征描述。基于此分析，我们对相应的生成流提供了精确描述，该流将基础高斯密度向前推至目标密度的近似。特别地，我们给出了生成混合分布均值与目标混合分布均值之间距离的闭式公式，并证明该距离以$\Theta_n(\frac{1}{n})$的速率衰减。最后，我们证明该速率实际上是贝叶斯最优的。