  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.
