  We present an extension of the cut-pursuit algorithm, introduced by Landrieu
and Obozinski (2017), to the graph total-variation regularization of functions
with a separable nondifferentiable part. We propose a modified algorithmic
scheme as well as adapted proofs of convergence. We also present a heuristic
approach for handling the cases in which the values associated to each vertex
of the graph are multidimensional. The performance of our algorithm, which we
demonstrate on difficult, ill-conditioned large-scale inverse and learning
problems, is such that it may in practice extend the scope of application of
the total-variation regularization.
