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.

中文翻译：

---

高维微分相关矩阵的推理

受基因组学中差异共表达分析的启发，我们在本文中考虑了高维差异相关矩阵的估计和测试。引入了自适应阈值程序并给出了理论保证。建立了最小最大收敛速率，并且表明所提出的估计器在具有近似稀疏差异的成对相关矩阵的集合上自适应地速率最优。仿真结果表明，该过程明显优于其他两种基于单独相关矩阵的单独估计的自然方法。该过程还通过对乳腺癌数据集的分析进行了说明，该数据集在基因共表达水平上提供了证据，即之前已经验证了其中一个子集的几个基因，与乳腺癌有关。还考虑了对微分相关矩阵的假设检验。介绍了一种特别适合针对稀疏替代品进行测试的测试。此外，还讨论了其他相关问题，包括单个稀疏相关矩阵的估计、微分协方差矩阵的估计和微分互相关矩阵的估计。