Large-scale Detection of Differential Sparsity Structure with FDR Control

Two-sample multiple testing has a wide range of applications. Most of the literature considers simultaneous tests of equality of parameters. This work takes a different perspective and investigates the null hypotheses that the two support sets are equal. This formulation of the testing problem is motivated by the fact that in many applications where the two parameter vectors being compared are both sparse, we might be more concerned about the detection of differential sparsity structures rather than the difference in parameter magnitudes. A general approach to problems of this type is developed via a novel double thresholding (DT) filter. The DT filter first constructs a sequence of pairs of ranking statistics that fulfill global symmetry properties, and then chooses two data-driven thresholds along the ranking to simultaneously control the false discovery rate (FDR) and maximize the number of rejections. Several applications of the methodology are given, including tests for large-scale correlation matrices, high-dimensional linear models and Gaussian graphical models.