Properties and refinements of the fused lasso

TL;DR

This paper analyzes the properties of the fused lasso and related estimators for recovering blocky and sparse signals from noisy data, proposing refinements and establishing conditions for accurate recovery and convergence rates.

Contribution

It introduces the fused adaptive lasso with improved properties and provides theoretical conditions and convergence rates for various sieve-based estimators.

Findings

Conditions for true block partition recovery

Conditions for true sparsity pattern recovery

Explicit convergence rates for sieve estimators

Abstract

We consider estimating an unknown signal, both blocky and sparse, which is corrupted by additive noise. We study three interrelated least squares procedures and their asymptotic properties. The first procedure is the fused lasso, put forward by Friedman et al. [Ann. Appl. Statist. 1 (2007) 302--332], which we modify into a different estimator, called the fused adaptive lasso, with better properties. The other two estimators we discuss solve least squares problems on sieves; one constrains the maximal $\ell_1$ norm and the maximal total variation seminorm, and the other restricts the number of blocks and the number of nonzero coordinates of the signal. We derive conditions for the recovery of the true block partition and the true sparsity patterns by the fused lasso and the fused adaptive lasso, and we derive convergence rates for the sieve estimators, explicitly in terms of the constraining parameters.