The Solution Path of the Generalized Lasso

We present a path algorithm for the generalized lasso problem, which penalizes the $\ell_1$ norm of a matrix $D$ times the coefficient vector. This setup applies to a wide range of useful problems, dictated by the choice of $D$. Our algorithm is based on solving the dual of the generalized lasso, an approach which greatly facilitates computation of the path, and reveals a nice representation for the solution. Using this representation, we develop an unbiased estimate of the degrees of freedom of the fit; for many specific choices of $D$, these estimates are quite intuitive. Our approach bears similarities to least angle regression (LARS), and when $D = I$, a simple modification to our method gives the LARS procedure exactly.