  Although Bayesian density estimation using discrete mixtures has good
performance in modest dimensions, there is a lack of statistical and
computational scalability to high-dimensional multivariate cases. To combat the
curse of dimensionality, it is necessary to assume the data are concentrated
near a lower-dimensional subspace. However, Bayesian methods for learning this
subspace along with the density of the data scale poorly computationally. To
solve this problem, we propose an empirical Bayes approach, which estimates a
multiscale dictionary using geometric multiresolution analysis in a first
stage. We use this dictionary within a multiscale mixture model, which allows
uncertainty in component allocation, mixture weights and scaling factors over a
binary tree. A computational algorithm is proposed, which scales efficiently to
massive dimensional problems. We provide some theoretical support for this
geometric density estimation (GEODE) method, and illustrate the performance
through simulated and real data examples.
