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

LASSO reloaded: a variational analysis perspective with applications to compressed sensing