Asymptotic distribution of constrained nearly‐isotonic graph fused lasso

Abstract This paper studies the asymptotic distribution of a constrained lasso‐type estimator for denoising signals defined on the nodes of a graph, where the underlying structure encodes relationships between variables. We show that, under suitable assumptions on the penalization parameters, the limiting distribution of the estimator is obtained by applying the corresponding constrained procedure to the asymptotic distribution of the unrestricted estimator. Thus, the constrained estimator shares the same convergence rate as the unrestricted estimator. Without the fusion penalty, the limiting distribution is obtained by applying the individual nearly isotonic estimators to the corresponding sub‐vectors of the unrestricted estimator's asymptotic distribution, similarly to the limit behavior of isotonic regression.