  Statistical inference from high-dimensional data with low-dimensional
structures has recently attracted lots of attention. In machine learning, deep
generative modeling approaches implicitly estimate distributions of complex
objects by creating new samples from the underlying distribution, and have
achieved great success in generating synthetic realistic-looking images and
texts. A key step in these approaches is the extraction of latent features or
representations (encoding) that can be used for accurately reconstructing the
original data (decoding). In other words, low-dimensional manifold structure is
implicitly assumed and utilized in the distribution modeling and estimation. To
understand the benefit of low-dimensional manifold structure in generative
modeling, we build a general minimax framework for distribution estimation on
unknown submanifold under adversarial losses, with suitable smoothness
assumptions on the target distribution and the manifold. The established
minimax rate elucidates how various problem characteristics, including
intrinsic dimensionality of the data and smoothness levels of the target
distribution and the manifold, affect the fundamental limit of high-dimensional
distribution estimation. To prove the minimax upper bound, we construct an
estimator based on a mixture of locally fitted generative models, which is
motivated by the partition of unity technique from differential geometry and is
necessary to cover cases where the underlying data manifold does not admit a
global parametrization. We also propose a data-driven adaptive estimator that
is shown to simultaneously attain within a logarithmic factor of the optimal
rate over a large collection of distribution classes.
