Consider the problem of minimizing the sum of two convex functions,
one being smooth and the other non-smooth. In this paper, we introduce a general
class of approximate proximal splitting (APS) methods for solving such minimization
problems. Methods in the APS class include many well-known algorithms
such as the proximal splitting method, the block coordinate descent method (BCD),
and the approximate gradient projection methods for smooth convex optimization.
We establish the linear convergence of APS methods under a local error bound
assumption. Since the latter is known to hold for compressive sensing and sparse
group LASSO problems, our analysis implies the linear convergence of the BCD
method for these problems without strong convexity assumption.