  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.
