  We consider Gaussian mixture models in high dimensions and concentrate on the
twin tasks of detection and feature selection. Under sparsity assumptions on
the difference in means, we derive information bounds and establish the
performance of various procedures, including the top sparse eigenvalue of the
sample covariance matrix and other projection tests based on moments, such as
the skewness and kurtosis tests of Malkovich and Afifi (1973), and other
variants which we were better able to control under the null.
