Nonparametric Kernel Estimation and Regression on Distributions

Low-dimensional embedding, manifold learning, clustering, classification, and anomaly detection are among the most important problems in machine learning. The existing methods usually consider the case when each instance has a fixed, finite-dimensional feature representation. We wish to expand the domain of consideration and let each instance correspond to a continuous probability distribution or a function from a nonparametric class in general. At times, these distributions are unknown, but we are given some i.i.d. samples from each distribution. Specifically, we wish to expand the domain of both covariates(inputs) and response(outputs) from real-valued pairs < Xi, Yi > to distributions.