This package is centered around computing the solution path of the generalized lasso problem, which minimizes the criterion $$ 1/2 \|y - X \beta\|_2^2 + \lambda \|D \beta\|_1. $$ The solution path is computed by solving the equivalent Lagrange dual problem. The dimension of the dual variable u is the number of rows of the penalty matrix D, and the primal (original) and dual solutions are related by $$ \hat{\beta} = y - D^T \hat{u} $$ when the predictor matrix X is the identity, and $$ \hat{\beta} = (X^T X)^{-1} (X^T y - D^T \hat{u}) $$ for a full column rank predictor matrix X. For column rank deficient matrices X, the solution path is not unique and not computed by this package. However, one can add a small ridge penalty to the above criterion, which can be re-expressed as a generalized lasso problem with full column rank predictor matrix X and hence yields a unique solution path.
Important use cases include the fused lasso, where D is the oriented incidence matrix of some underlying graph (the orientations being arbitrary), and trend filtering, where D is the discrete difference operator of any given order k.
The general function genlasso computes a solution path
for any penalty matrix D and full column rank predictor matrix X
(adding a ridge penalty when X is rank deficient). For the fused lasso
and trend filtering problems, the specialty functions
fusedlasso and trendfilter should be used
as they deliver a significant increase in speed and numerical
stability.
For a walk-through of using the package for statistical modelling see the included package vignette; for the appropriate background material see the generalized lasso paper referenced below.
Taylor B. Arnold and Ryan J. Tibshirani
Tibshirani, R. J. and Taylor, J. (2011), "The solution path of the generalized lasso", Annals of Statistics 39 (3) 1335--1371.
Arnold, T. B. and Tibshirani, R. J. (2014), "Efficient implementations of the generalized lasso dual path algorithm", arXiv: 1405.3222.
genlasso, fusedlasso,
trendfilter