The proximal splitting algorithms can iteratively approximate an unspecial vector among possibly infinitely many minimizers of a superposition of multiple nonsmooth convex functions. With elegant translations of the solution set, i.e., the set of all minimizers, into the fixed point sets of nonexpansive mappings, the hybrid steepest descent method allows further strategic selection of a most desirable vector among the solution set, by minimizing an additional convex function over the solution set. In this talk, we introduce fixed point theoretic interpretations of variety of proximal splitting algorithms and their enhancements by the hybrid steepest descent method with applications to recent advanced statistical estimation problems.