  L1 -penalized regression methods such as the Lasso (Tibshirani 1996) that
achieve both variable selection and shrinkage have been very popular. An
extension of this method is the Fused Lasso (Tibshirani and Wang 2007), which
allows for the incorporation of external information into the model. In this
article, we develop new and fast algorithms for solving the Fused Lasso which
are based on coordinate-wise optimization. This class of algorithms has
recently been applied very successfully to solve L1 -penalized problems very
quickly (Friedman et al. 2007). As a straightforward coordinate-wise procedure
does not converge to the global optimum in general, we adapt it in two ways,
using maximum-flow algorithms and a Huber penalty based approximation to the
loss function. In a simulation study, we evaluate the speed of these algorithms
and compare them to other standard methods. As the Huber-penalty based method
is only approximate, we also evaluate its accuracy. Apart from this, we also
extend the Fused Lasso to logistic as well as proportional hazards models and
allow for a more flexible penalty structure.
