Title: Some exercises with the Lasso and its compatibility constant,
Abstract: We consider the Lasso for a noiseless experiment where one has observations
$X \beta^0$ and uses the penalized version of basis pursuit. We compute for
some special designs the compatibility constant, a quantity closely related to
the restricted eigenvalue. We moreover show the dependence of the (penalized)
prediction error on this compatibility constant. This exercise illustrates that
compatibility is necessarily entering into the bounds for the (penalized)
prediction error and that the bounds in the literature therefore are - up to
constants - tight. We also give conditions that show that in the noisy case the
dominating term for the prediction error is given by the prediction error of
the noiseless case.