Nonconvex Sparse Regularization and Splitting Algorithms

Nonconvex regularization functions such as the l(p) quasinorm (0 < p < 1) can recover sparser solutions from fewer measurements than the convex l(1) regularization function. They have been widely used for compressive sensing and signal processing. This chapter briefly reviews the development of algorithms for nonconvex regularization. Because nonconvex regularization usually has different regularity properties from other functions in a problem, we often apply operator splitting (forward-backward splitting) to develop algorithms that treat them separately. The treatment on nonconvex regularization is via the proximal mapping. We also review another class of coordinate descent algorithms that work for both convex and nonconvex functions. They split variables into small, possibly parallel, subproblems, each of which updates a variable while fixing others. Their theory and applications have been recently extended to cover nonconvex regularization functions, which we review in this chapter. Finally, we also briefly mention an ADMM-based algorithm for nonconvex regularization, as well as the recent algorithms for the so-called nonconvex sort l(1) and l(1)-l(2) minimization.