Principle components analysis (PCA) is an important technique for reducing the
dimensionality of data, and is often used as a preprocessing step in machine
learning algorithms.  We propose a algorithm that produces an approximation to
the top-k PCA subspace of a collection of data points while guaranteeing
differential privacy.  We provide theoretical upper and lower bounds when k =
1 on the number of points needed to guarantee a given privacy level and
approximation error.  We show that our algorithm is nearly is nearly optimal
in terms of scaling with the data dimension.  We demonstrate our method on a
number of data sets and indicate a number of promising future directions.  This
is joint work with Kamalika Chaudhuri and Kaushik Sinha.