Variable selection, monotone likelihood ratio and group sparsity

In the pivotal variable selection problem, we derive exact nonasymptotic minimax selector over class of all s-sparse vectors, which is also Bayes with respect to uniform prior. While this optimal is, in general, not realizable polynomial time, show that its tractable counterpart (the scan selector) attains expected Hamming risk within factor 2, and probability wrong recovery. As a consequence, establish explicit lower bounds under monotone likelihood ratio property obtain tight characterization terms best separable risk. We apply these general results necessary sufficient conditions almost full recovery location model light tail distributions problem group Gaussian noise more anisotropic sub-Gaussian noise. Numerical illustrate our theoretical findings.