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Low-dimensional embedding, manifold learning, clustering, classification, andanomaly 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 differentsetting. We assume that each instance corresponds to a continuous probabilitydistribution. These distributions are unknown, but we are given some i.i.d.samples from each distribution. Our goal is to estimate the distances betweenthese distributions and use these distances to perform low-dimensionalembedding, clustering/classification, or anomaly detection for thedistributions. We present estimation algorithms, describe how to apply them formachine learning tasks on distributions, and show empirical results onsynthetic data, real word images, and astronomical data sets.

Nonparametric Divergence Estimation with Applications to Machine Learning on Distributions