On the LASSO and its dual

Proposed by Tibshirani, the least absolute shrinkage and selection operator (LASSO) estimates a vector of regression coefficients by minimizing the re sidual sum of squares subject to a constraint on the l(1)-norm of the coeff icient vector. The LASSO estimator typically has one or more zero elements and thus shares characteristics of both shrinkage estimation and variable s election. In this article we treat the LASSO as a convex programming proble m and derive its dual. Consideration of the primal and dual problems togeth er leads to important new insights into the characteristics of the LASSO es timator and to an improved method for estimating its covariance matrix. Usi ng these results we also develop an efficient algorithm for computing LASSO estimates which is usable even in cases where the number of regressors exc eeds the number of observations. An S-Plus library based on this algorithm is available from StatLib.