  We consider testing the equality of two high-dimensional covariance matrices
by carrying out a multi-level thresholding procedure, which is designed to
detect sparse and faint differences between the covariances. A novel
U-statistic composition is developed to establish the asymptotic distribution
of the thresholding statistics in conjunction with the matrix blocking and the
coupling techniques. We propose a multi-thresholding test that is shown to be
powerful in detecting sparse and weak differences between two covariance
matrices. The test is shown to have attractive detection boundary and to attain
the optimal minimax rate in the signal strength under different regimes of high
dimensionality and the sparsity of the signal. Simulation studies are conducted
to demonstrate the utility of the proposed test.
