Denoising and change point localisation in piecewise-constant high-dimensional regression coefficients

TL;DR

This paper analyzes the theoretical properties of the fused lasso in high-dimensional linear regression with sparse, piecewise-constant coefficients, introducing a new restricted isometry condition and a post-processing method.

Contribution

It introduces a novel restricted isometry condition for the design matrix and provides estimation bounds for fused lasso in high-dimensional settings with sparse, piecewise-constant coefficients.

Findings

Estimation error depends on either lasso or fused lasso rate.

The proposed post-processing recovers the piecewise-constant pattern.

Numerical experiments validate theoretical bounds.

Abstract

We study the theoretical properties of the fused lasso procedure originally proposed by \cite{tibshirani2005sparsity} in the context of a linear regression model in which the regression coefficient are totally ordered and assumed to be sparse and piecewise constant. Despite its popularity, to the best of our knowledge, estimation error bounds in high-dimensional settings have only been obtained for the simple case in which the design matrix is the identity matrix. We formulate a novel restricted isometry condition on the design matrix that is tailored to the fused lasso estimator and derive estimation bounds for both the constrained version of the fused lasso assuming dense coefficients and for its penalised version. We observe that the estimation error can be dominated by either the lasso or the fused lasso rate, depending on whether the number of non-zero coefficient is larger than the number of piece-wise constant segments. Finally, we devise a post-processing procedure to recover the piecewise-constant pattern of the coefficients. Extensive numerical experiments support our theoretical findings.