We propose a Monte-Carlo-based method for reconstructing sparse signals in the formulation of sparse linear regression in a high-dimensional setting. The basic idea of this algorithm is to explicitly select variables or covariates to represent a given data vector or responses and accept randomly generated updates of that selection if and only if the energy or cost function decreases. This algorithm is called the greedy Monte-Carlo (GMC) search algorithm. Its performance is examined via numerical experiments, which suggests that in the noiseless case, GMC can achieve perfect reconstruction in undersampling situations of a reasonable level: it can outperform the $\ell_1$ relaxation but does not reach the algorithmic limit of MC-based methods theoretically clarified by an earlier analysis. Additionally, an experiment on a real-world dataset supports the practicality of GMC.

中文翻译：

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通过贪心蒙特卡罗搜索重建稀疏信号

我们提出了一种基于蒙特卡罗的方法，用于在高维设置的稀疏线性回归公式中重建稀疏信号。该算法的基本思想是明确选择变量或协变量来表示给定的数据向量或响应，并当且仅当能量或成本函数减小时接受该选择的随机生成的更新。这种算法称为贪婪蒙特卡罗（GMC）搜索算法。通过数值实验对其性能进行了检验，这表明在无噪声情况下，GMC 可以在合理水平的欠采样情况下实现完美的重建：它可以胜过 $\ell_1$ 松弛，但没有达到基于 MC 的方法的算法限制之前的分析在理论上澄清了这一点。此外，