  The Lasso is a popular regression method for high-dimensional problems in
which the number of parameters $\theta_1,\dots,\theta_N$, is larger than the
number $n$ of samples: $N>n$. A useful heuristics relates the statistical
properties of the Lasso estimator to that of a simple soft-thresholding
denoiser,in a denoising problem in which the parameters $(\theta_i)_{i\le N}$
are observed in Gaussian noise, with a carefully tuned variance. Earlier work
confirmed this picture in the limit $n,N\to\infty$, pointwise in the parameters
$\theta$, and in the value of the regularization parameter.
  Here, we consider a standard random design model and prove exponential
concentration of its empirical distribution around the prediction provided by
the Gaussian denoising model. Crucially, our results are uniform with respect
to $\theta$ belonging to $\ell_q$ balls, $q\in [0,1]$, and with respect to the
regularization parameter. This allows to derive sharp results for the
performances of various data-driven procedures to tune the regularization.
  Our proofs make use of Gaussian comparison inequalities, and in particular of
a version of Gordon's minimax theorem developed by Thrampoulidis, Oymak, and
Hassibi, which controls the optimum value of the Lasso optimization problem.
Crucially, we prove a stability property of the minimizer in Wasserstein
distance, that allows to characterize properties of the minimizer itself.
