The Degrees of Freedom of Partly Smooth Regularizers

In this paper, we are concerned with regularized regression problems where the prior penalty is a piecewise regular/partly smooth gauge whose active manifold is linear. This encompasses as special cases the Lasso ($\lun$ regularizer), the group Lasso ($\lun-\ldeux$ regularizer) and the $\linf$-norm regularizer penalties. This also includes so-called analysis-type priors, i.e. composition of the previously mentioned functionals with linear operators, a typical example being the total variation prior. We study the sensitivity of {\textit{any}} regularized minimizer to perturbations of the observations and provide its precise local parameterization. Our main result shows that, when the observations are outside a set of zero Lebesgue measure, the predictor moves locally stably along the same linear space as the observations undergo small perturbations. This local stability is a consequence of the piecewise regularity of the gauge, which in turn plays a pivotal role to get a closed form expression for the variations of the predictor w.r.t. observations which holds almost everywhere. When the perturbation is random (with an appropriate continuous distribution), this allows us to derive an unbiased estimator of the degrees of freedom and of the risk of the estimator prediction. Our results hold true without placing any assumption on the design matrix, should it be full column rank or not. They generalize those already known in the literature such as the Lasso problem, the general Lasso problem (analysis $\lun$-penalty), or the group Lasso where existing results for the latter assume that the design is full column rank.