  Motivated by differential co-expression analysis in genomics, we consider in
this paper estimation and testing of high-dimensional differential correlation
matrices. An adaptive thresholding procedure is introduced and theoretical
guarantees are given. Minimax rate of convergence is established and the
proposed estimator is shown to be adaptively rate-optimal over collections of
paired correlation matrices with approximately sparse differences. Simulation
results show that the procedure significantly outperforms two other natural
methods that are based on separate estimation of the individual correlation
matrices. The procedure is also illustrated through an analysis of a breast
cancer dataset, which provides evidence at the gene co-expression level that
several genes, of which a subset has been previously verified, are associated
with the breast cancer. Hypothesis testing on the differential correlation
matrices is also considered. A test, which is particularly well suited for
testing against sparse alternatives, is introduced. In addition, other related
problems, including estimation of a single sparse correlation matrix,
estimation of the differential covariance matrices, and estimation of the
differential cross-correlation matrices, are also discussed.
