Asymptotic Distribution of Constrained Nearly-Isotonic Graph Fused Lasso

TL;DR

This paper derives the asymptotic distribution of a constrained graph-based fused lasso estimator for signal denoising, showing it shares the same convergence rate as the unrestricted estimator under certain conditions.

Contribution

It provides a theoretical analysis of the limiting distribution of constrained nearly-isotonic graph fused lasso estimators, extending understanding of their asymptotic behavior.

Findings

Constrained estimator's limiting distribution is derived from the unrestricted estimator's distribution.

Under certain assumptions, the constrained estimator has the same convergence rate as the unrestricted one.

Without the fusion penalty, the limiting distribution relates to individual nearly isotonic estimators.

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 individual nearly isotonic estimators to the corresponding sub-vectors of the unrestricted estimator's asymptotic distribution, similarly to the limiting behavior of isotonic regression.