  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.
