In the context of high-dimensional linear regression models, we propose an algorithm of exact support recovery in the setting of noisy compressed sensing where all entries of the design matrix are independent and identically distributed standard Gaussian. This algorithm achieves the same conditions of exact recovery as the exhaustive search (maximal likelihood) decoder, and has an advantage over the latter of being adaptive to all parameters of the problem and computable in polynomial time. The core of our analysis consists in the study of the non-asymptotic minimax Hamming risk of variable selection. This allows us to derive a procedure, which is nearly optimal in a non-asymptotic minimax sense. Then, we develop its adaptive version, and propose a robust variant of the method to handle datasets with outliers and heavy-tailed distributions of observations. The resulting polynomial time procedure is near optimal, adaptive to all parameters of the problem and also robust.

中文翻译：

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最优变量选择和自适应噪声压缩感知

在高维线性回归模型的背景下，我们提出了一种在噪声压缩感知设置中精确支持恢复的算法，其中设计矩阵的所有条目都是独立且同分布的标准高斯。该算法实现了与穷举搜索（最大似然）解码器相同的精确恢复条件，并且比后者具有适应问题的所有参数并且在多项式时间内可计算的优势。我们分析的核心在于研究变量选择的非渐近极小极大汉明风险。这允许我们推导出一个过程，它在非渐近极大极小意义上几乎是最优的。然后，我们开发了它的自适应版本，并提出了一种强大的方法变体来处理具有异常值和重尾观测分布的数据集。由此产生的多项式时间过程接近最优，适应问题的所有参数，而且还很健壮。