This paper provides a variational analysis of the unconstrained formulation of the LASSO problem, ubiquitous in statistical learning, signal processing, and inverse problems. In particular, we establish smoothness results for the optimal value as well as Lipschitz properties of the optimal solution as functions of the right-hand side (or measurement vector) and the regularization parameter. Moreover, we show how to apply the proposed variational analysis to study the sensitivity of the optimal solution to the tuning parameter in the context of compressed sensing with subgaussian measurements. Our theoretical findings are validated by numerical experiments.

翻译：本文对LASSO问题的无约束形式进行了变分分析，该问题在统计学习、信号处理和反问题中普遍存在。我们特别建立了最优值的光滑性结果，以及最优解作为右端项（或测量向量）和正则化参数函数的Lipschitz性质。此外，我们展示了如何将所提出的变分分析应用于研究亚高斯测量下压缩感知场景中优化解对调优参数的敏感性。我们的理论发现通过数值实验得到了验证。