  We consider the problem of multivariate density estimation when the unknown
density is assumed to follow a particular form of dimensionality reduction, a
noisy independent factor analysis (IFA) model. In this model the data are
generated by a number of latent independent components having unknown
distributions and are observed in Gaussian noise. We do not assume that either
the number of components or the matrix mixing the components are known. We show
that the densities of this form can be estimated with a fast rate. Using the
mirror averaging aggregation algorithm, we construct a density estimator which
achieves a nearly parametric rate log^(1/4)n/sqrt(n), independent of the
dimensionality of the data, as the sample size $n$ tends to infinity. This
estimator is adaptive to the number of components, their distributions and the
mixing matrix. We then apply this density estimator to construct nonparametric
plug-in classifiers and show that they achieve the best obtainable rate of the
excess Bayes risk, to within a logarithmic factor independent of the dimension
of the data. Applications of this classifier to simulated data sets and to real
data from a remote sensing experiment show promising results.
