Variable selection, monotone likelihood ratio and group sparsity
研究了变量选择问题中的最优选择器，推导了精确的非渐近极小化风险，并给出了可计算的选择器（扫描选择器）在风险上达到因子2内的最优性，同时应用于位置模型和组变量选择问题。
In the pivotal variable selection problem, we derive the exact nonasymptotic minimax selector over the class of all s-sparse vectors, which is also the Bayes selector with respect to the uniform prior. While this optimal selector is, in general, not realizable in polynomial time, we show that its tractable counterpart (the scan selector) attains the minimax expected Hamming risk to within factor 2, and is also exact minimax with respect to the probability of wrong recovery. As a consequence, we establish explicit lower bounds under the monotone likelihood ratio property and we obtain a tight characterization of the minimax risk in terms of the best separable selector risk. We apply these general results to derive necessary and sufficient conditions of exact and almost full recovery in the location model with light tail distributions and in the problem of group variable selection under Gaussian noise and under more general anisotropic sub-Gaussian noise. Numerical results illustrate our theoretical findings.