Principal components analysis (PCA) is a standard tool for identifying good
low-dimensional approximations to data in high dimension. Many data sets of
interest contain private or sensitive information about individuals. Algorithms
which operate on such data should be sensitive to the privacy risks in
publishing their outputs. Differential privacy is a framework for developing
tradeoffs between privacy and the utility of these outputs. In this paper we
investigate the theory and empirical performance of differentially private
approximations to PCA and propose a new method which explicitly optimizes the
utility of the output. We show that the sample complexity of the proposed
method differs from the existing procedure in the scaling with the data
dimension, and that our method is nearly optimal in terms of this scaling. We
furthermore illustrate our results, showing that on real data there is a large
performance gap between the existing method and our method.Comment: 37 pages, 8 figures; final version to appear in the Journal of
Machine Learning Research, preliminary version was at NIPS 201