  Despite the fact that generative models are extremely successful in practice,
the theory underlying this phenomenon is only starting to catch up with
practice. In this work we address the question of the universality of
generative models: is it true that neural networks can approximate any data
manifold arbitrarily well? We provide a positive answer to this question and
show that under mild assumptions on the activation function one can always find
a feedforward neural network that maps the latent space onto a set located
within the specified Hausdorff distance from the desired data manifold. We also
prove similar theorems for the case of multiclass generative models and cycle
generative models, trained to map samples from one manifold to another and vice
versa.
