  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. Here we consider a different
setting. We assume that each instance corresponds to a continuous probability
distribution. These distributions are unknown, but we are given some i.i.d.
samples from each distribution. Our goal is to estimate the distances between
these distributions and use these distances to perform low-dimensional
embedding, clustering/classification, or anomaly detection for the
distributions. We present estimation algorithms, describe how to apply them for
machine learning tasks on distributions, and show empirical results on
synthetic data, real word images, and astronomical data sets.
