the optimal solution for the bottom three panels only is regularizedtherehorizontal lines indicate vectors [ ]for which | | * lasso the problem of ridge regression discussed in example boils down to solving problem of the form in ( )involving squared -norm penalty kbk natural question to ask is whether we can replace the squared -norm penalty by different penalty term replacing it with -norm gives the lasso (least absolute shrinkage and selection operatorthe lasso equivalent of the ridge regression problem ( is thus xb kbk , min where kbk ( pp = |bi this is again convex optimization problem unlike ridge regressionthe lasso generally does not have an explicit solutionand so numerical methods must be used to solve it note that the problem ( is of the form min , (xg(zsubject to ax bz ( with :[ ] :ba :[ ] :- and : (vector of zeros)and convex functions ( : [ xx and ( :gkzk there exist efficient algorithms for solving such problemsincluding the alternating direction method of multipliers (admm[ we refer to example ?for details on this algorithm we repeat the examples from figure but now using lasso regression and taking the square roots of the previous regularization parameters the results are displayed in figure