A generalized lasso formulation of weighted and regularized (nearly-)isotonic regression on graphs

Abstract

In this paper we study a regularized version of nearly-isotonic regression with respect to a general partial order. In this estimator the sum of squares is weighted, with some of the weights allowed to be zero, and an additional quadratic penalty is imposed. The only assumption we make is that the quadratic part of the objective function is strictly convex, and we show that this assumption is mild. First, we show that for all sufficiently large values of the penalty parameter the estimator coincides with the regularized isotonic regression, in which the penalty is replaced by the isotonic constraint. Further, we prove that the resulting problem can be written as a generalized lasso problem with a design matrix of full column rank and with the penalty matrix unchanged. Next, we derive the dual problem, which is a quadratic program with box constraints, and write it in a sparse form that can be solved by standard quadratic programming solvers for sparse problems.