Constrained grouped sparsity

In this paper, we propose an algorithm encouraging groupsparsity under some convex constraint. It stems from some applicationswhere the regression coefficients are subject to constraints, for examplenonnegativity and the explanatory variables are not suitable to beorthogonalized within groups. It takes the form of the group LASSObased on linear regression model where a L1/L2 norm is imposed ongroup coefficients to achieve group sparsity. It differs from the originalgroup LASSO in the following ways. First, the regression coefficientsmust obey some convex constraints. Second, there is no requirement fororthogonality of the variables within individual groups. For these reasons,the simple blockwise coordinate descent for all group coefficients isno longer applicable and a special treatment for the constraint is necessary.The algorithm we proposed in this paper is an alternating directionmethod, and both exact and inexact solutions are provided. The inexactversion simplifies the computation while retaining practical convergence.As an approximation to group L0, it can be applied to data analysiswhere group fitting is essential and the coefficients are constrained. Itmay serve as a screening procedure to reduce the number of the groupswhen the number of total groups is prohibitively high.