Recent Developments of Nonparametric Methods and Their Applications in Data Science and AI

Generative models based on latent variables, such as generative adversarial networks (GANs) and variational auto-encoders (VAEs), have gained lots of interests due to their impressive performance in many fields. However, many data such as natural images usually do not populate the ambient Euclidean space but instead reside in a lower-dimensional manifold. Thus, an inappropriate choice of the latent dimension fails to uncover the structure of the data, possibly resulting in mismatch of latent representations and poor generative qualities. Toward addressing these problems, we propose a novel framework called the latent Wasserstein GAN (LWGAN) that fuses the Wasserstein auto-encoder and the Wasserstein GAN so that the intrinsic dimension of the data manifold can be adaptively learned by a modified informative latent distribution. We prove that there exist an encoder network and a generator network in such a way that the intrinsic dimension of the learned encoding distribution is equal to the dimension of the data manifold. We theoretically establish that our estimated intrinsic dimension is a consistent estimate of the true dimension of the data manifold. Meanwhile, we provide an upper bound on the generalization error of LWGAN, implying that we force the synthetic data distribution to be similar to the real data distribution from a population perspective. Comprehensive empirical experiments verify our framework and show that LWGAN is able to identify the correct intrinsic dimension under several scenarios, and simultaneously generate high-quality synthetic data by sampling from the learned latent distribution. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.  

Existing methods for spatial data either have difficulty capturing heterogeneous patterns over complex domains or overlook the heterogeneity in the tail of the response distribution. We introduce a flexible quantile spatial model framework, which can simultaneously capture spatial nonstationarity and heterogeneity via constant and spatially varying coefficients. It also allows researchers to study patterns across different tails of the response distribution. We propose a smoothed quantile bivariate triangulation method based on penalized splines on triangulation and convolution smoothing in the quantile loss. The developed method can effectively capture spatial nonstationarity while preserving critical data features such as shape and smoothness across complex and irregular domains. Under some regularity conditions, we show that the proposed estimator can achieve an optimal convergence rate under the L2-norm. In addition, we establish the Bahadur representation of the estimator, which allows us to establish the asymptotic normality for the constant coefficient estimator and construct asymptotic confidence intervals. To improve finite-sample performance, we also consider a wild bootstrap method for constructing confidence intervals. Through simulation studies, we demonstrate the numerical and computational advantages of the proposed estimator over existing methods. The application of the proposed method to study the spatial heterogeneity of US mortality demonstrates that the mortality rates depend on socioeconomic factors differently across space and the tails of the mortality distribution. This is joint work with PhD student Jilei Lin and his advisor, Dr. Judy Wang, at The George Washington University, as well as Dr. Myungjin Kim from Kyungpook National University.