Nonparametric Distribution Regression and Change Point Detection in High-Dimensions

This dissertation investigates two high-dimensional problems—change-point analysis in functional linear regression and nonparametric density estimation using modern generative models. First, we introduce the Functional Regression Binary Segmentation (FRBS) algorithm for change-point detection in the slope function when predictors are functions and responses are scalars. This algorithm utilizes the predictive power of piece-wise constant functional linear regression models in the reproducing kernel Hilbert space framework. We further propose a refinement step that improves the localization rate of the initial estimator output by FRBS, and derive asymptotic distributions of the refined estimators for two different regimes determined by the magnitude of a change. Second, we study the theoretical properties of conditional deep generative models under the statistical framework of distribution regression where the response variable lies in a high-dimensional ambient space but concentrates around a potentially lower-dimensional manifold. Our results lead to the convergence rate of a sieve maximum likelihood estimator (MLE) for estimating the conditional distribution (and its devolved counterpart) of the response given predictors in the Hellinger (Wasserstein) metric. Third, we provide the first finite-sample theoretical analysis for flow matching—a simulation-free generative modeling approach. The velocity field is estimated via empirical risk minimization over a suitably designed ReLU network class. A switching argument is used to transfer the estimation error from the learned velocity field and time discretization to the resulting transport map, yielding an overall Wasserstein-2 error bound.<p></p>