  This work studies formal utility and privacy guarantees for a simple
multiplicative database transformation, where the data are compressed by a
random linear or affine transformation, reducing the number of data records
substantially, while preserving the number of original input variables. We
provide an analysis framework inspired by a recent concept known as
differential privacy (Dwork 06). Our goal is to show that, despite the general
difficulty of achieving the differential privacy guarantee, it is possible to
publish synthetic data that are useful for a number of common statistical
learning applications. This includes high dimensional sparse regression (Zhou
et al. 07), principal component analysis (PCA), and other statistical measures
(Liu et al. 06) based on the covariance of the initial data.
