  It is now practically the norm for data to be very high dimensional in areas
such as genetics, machine vision, image analysis and many others. When
analyzing such data, parametric models are often too inflexible while
nonparametric procedures tend to be non-robust because of insufficient data on
these high dimensional spaces. It is often the case with high-dimensional data
that most of the variability tends to be along a few directions, or more
generally along a much smaller dimensional submanifold of the data space. In
this article, we propose a class of models that flexibly learn about this
submanifold and its dimension which simultaneously performs dimension
reduction. As a result, density estimation is carried out efficiently. When
performing classification with a large predictor space, our approach allows the
category probabilities to vary nonparametrically with a few features expressed
as linear combinations of the predictors. As opposed to many black-box methods
for dimensionality reduction, the proposed model is appealing in having clearly
interpretable and identifiable parameters. Gibbs sampling methods are developed
for posterior computation, and the methods are illustrated in simulated and
real data applications.
