This paper studies well-posedness and parameter sensitivity of the Square Root LASSO (SR-LASSO), an optimization model for recovering sparse solutions to linear inverse problems in finite dimension. An advantage of the SR-LASSO (e.g., over the standard LASSO) is that the optimal tuning of the regularization parameter is robust with respect to measurement noise. This paper provides three point-based regularity conditions at a solution of the SR-LASSO: the weak, intermediate, and strong assumptions. It is shown that the weak assumption implies uniqueness of the solution in question. The intermediate assumption yields a directionally differentiable and locally Lipschitz solution map (with explicit Lipschitz bounds), whereas the strong assumption gives continuous differentiability of said map around the point in question. Our analysis leads to new theoretical insights on the comparison between SR-LASSO and LASSO from the viewpoint of tuning parameter sensitivity: noise-robust optimal parameter choice for SR-LASSO comes at the "price" of elevated tuning parameter sensitivity. Numerical results support and showcase the theoretical findings.

翻译：本文研究平方根LASSO（SR-LASSO）的适定性和参数敏感性，该模型用于在有限维空间中求解线性逆问题的稀疏解。相较于标准LASSO，SR-LASSO的一个优势在于正则化参数的最优调参对测量噪声具有鲁棒性。本文在SR-LASSO解处提出了三种基于点的正则性条件：弱假设、中等假设和强假设。研究表明，弱假设可确保所讨论解的唯一性；中等假设能导出方向可微且局部Lipschitz连续的解映射（含显式Lipschitz界）；而强假设则保证该映射在目标点附近具有连续可微性。我们的分析从调参敏感性视角揭示了SR-LASSO与LASSO比较的新理论见解：SR-LASSO在噪声环境下实现最优参数选择的鲁棒性，是以“牺牲”更高的调参敏感性为代价。数值实验结果验证并展示了上述理论发现。