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问题不受限制的表述、统计学习、信号处理和反面问题无处不在、对LASSO问题无限制的表述进行了不同分析。特别是,我们为最佳解决方案的最佳价值和利普施奇茨特性确定顺畅的结果,作为右侧(或测量矢量)和正规化参数的功能。此外,我们展示了如何应用拟议的变异分析来研究最佳解决方案在采用亚高巴苏西测量法的压缩遥感条件下对调控参数的敏感性。我们的理论结论通过数字实验得到验证。