Properties and refinements of the fused Lasso. (English) Zbl 1173.62027

Summary: 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 J. Friedman et al. [Ann. Appl. Stat. 1, No. 2, 302–332 (2007; Zbl 1378.90064)], 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 non-zero 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.


62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)


Zbl 1378.90064


Full Text: DOI arXiv


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