Properties and Refinements of The Fused Lasso

We consider estimating an unknown signal, which is both blocky and sparse, 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. (2007), 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 constraining the maximal `1 norm and the maximal total variation seminorm, the other restricting the number of blocks and the number of 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 convergence rates for the sieve estimators, explicitly in terms of the constraining parameters.