Denoising and Change Point Localisation in Piecewise-Constant High-Dimensional Regression Coefficients

• Activity Number: 75 - Invited EPoster Session II
• Type: Invited
• Date/Time: Sunday, August 7, 2022 : 9:35 PM to 10:30 PM
• Sponsor: Section on Statistical Learning and Data Science
• Author(s): OSCAR HERNAN MADRID PADILLA* and Fan Wang and Yi Yu and Alessandro Rinaldo
• Companies: University of California, Los Angeles and University of Warwick and University of Warwick and Carnegie Mellon University
• Keywords: Total variation; denosing; change point detection

Abstract

We study the theoretical properties of the fused lasso procedure originally proposed by Tibshirani et al. (2005) in the context of a linear regression model in which the regression coefficients 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.