  This paper generalizes the Maurer--Pontil framework of finite-dimensional
lossy coding schemes to the setting where a high-dimensional random vector is
mapped to an element of a compact set of latent representations in a
lower-dimensional Euclidean space, and the reconstruction map belongs to a
given class of nonlinear maps. Under this setup, which encompasses a broad
class of unsupervised representation learning problems, we establish a
connection to approximate generative modeling under structural constraints
using the tools from the theory of optimal transportation. Next, we consider
problem of learning a coding scheme on the basis of a finite collection of
training samples and present generalization bounds that hold with high
probability. We then illustrate the general theory in the setting where the
reconstruction maps are implemented by deep neural nets.
