  In this paper, we study the local linear convergence properties of a
versatile class of Primal-Dual splitting methods for minimizing composite
non-smooth convex optimization problems. Under the assumption that the
non-smooth components of the problem are partly smooth relative to smooth
manifolds, we present a unified local convergence analysis framework for these
methods. More precisely, in our framework we first show that (i) the sequences
generated by Primal-Dual splitting methods identify a pair of primal and dual
smooth manifolds in a finite number of iterations, and then (ii) enter a local
linear convergence regime, which is characterized based on the structure of the
underlying active smooth manifolds. We also show how our results for
Primal-Dual splitting can be specialized to cover existing ones on
Forward-Backward splitting and Douglas-Rachford splitting/ADMM (alternating
direction methods of multipliers). Moreover, based on these obtained local
convergence analysis result, several practical acceleration techniques are
discussed. To exemplify the usefulness of the obtained result, we consider
several concrete numerical experiments arising from fields including
signal/image processing, inverse problems and machine learning, etc. The
demonstration not only verifies the local linear convergence behaviour of
Primal-Dual splitting methods, but also the insights on how to accelerate them
in practice.
