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.

本文研究了三个最小二乘程序及其渐近特性，提出了一种名为融合自适应 Lasso 的估计器，讨论了节省筛中 L1 范数和变化的最大总变差半范数，以及限制实现信号阻塞的数量和非零坐标的数量的另外两个估计量，推导了融合 lasso 和融合自适应 lasso 恢复真实块分区和稀疏模式的条件，并明确了筛选器的收敛速度与限良参数的关系。