Title: Deep generative models for nonparametric estimation of a singular distribution
Authors: Minwoo Chae - Pohang University of Science and Technology (Korea, South) [presenting]
Lizhen Lin - The University of Notre Dame (United States)
Yongdai Kim - Seoul National University (Korea, South)
Dongha Kim - Seoul National University (Korea, South)
Abstract: While deep generative models are popularly used to model high-dimensional data, there is a lack of mathematical understanding. We investigate the statistical properties of deep generative models from a nonparametric distribution estimation viewpoint. In the considered model, data are assumed to concentrate around some low-dimensional structure. Estimating the distribution supported on this low-dimensional structure, such as a low-dimensional manifold, is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. In particular, a likelihood approach can fail to estimate the target distribution consistently. We obtain convergence rates with respect to the Wasserstein metric for two methods: a sieve MLE based on the perturbed data and a GAN type estimator. Our analysis gives some insights into i) how deep generative models can avoid the curse of dimensionality, ii) how likelihood approaches work for singular distribution estimation, and iii) why GAN performs better than likelihood approaches.