Department of Statistics Unitmark
Dietrich College of Humanities and Social Sciences

Degrees of Freedom in Lasso Problems

Publication Date

November, 2011

Publication Type

Tech Report


Ryan J. Tibshirani and Jonathan Taylor


We derive the degrees of freedom of the lasso fit, placing no assumptions on the predictor matrix X. Like the well-known result of Zou, Hastie and Tibshirani [Ann. Statist. 35 (2007) 2173-2192], which gives the degrees of freedom of the lasso fit when X has full column rank, we express our result in terms of the active set of a lasso solution. We extend this result to cover the degrees of freedom of the generalized lasso fit for an arbitrary predictor matrix X (and an arbitrary penalty matrix D). Though our focus is degrees of freedom, we establish some intermediate results on the lasso and generalized lasso that may be interesting on their own.