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In this paper, we investigate the degrees of freedom ($\dof$) of penalized$\ell_1$ minimization (also known as the Lasso) for linear regression models.We give a closed-form expression of the $\dof$ of the Lasso response. Namely,we show that for any given Lasso regularization parameter $\lambda$ and anyobserved data $y$ belonging to a set of full (Lebesgue) measure, thecardinality of the support of a particular solution of the Lasso problem is anunbiased estimator of the degrees of freedom. This is achieved without the needof uniqueness of the Lasso solution. Thus, our result holds true for both theunderdetermined and the overdetermined case, where the latter was originallystudied in \cite{zou}. We also show, by providing a simple counterexample, thatalthough the $\dof$ theorem of \cite{zou} is correct, their proof contains aflaw since their divergence formula holds on a different set of a full measurethan the one that they claim. An effective estimator of the number of degreesof freedom may have several applications including an objectively guided choiceof the regularization parameter in the Lasso through the $\sure$ framework. Ourtheoretical findings are illustrated through several numerical simulations.

The degrees of freedom of the Lasso for general design matrix