A Unified Primal Dual Active Set Algorithm for Nonconvex Sparse Recovery
This addresses sparse recovery for signal processing or statistics, offering a novel algorithm with strong performance gains, though it is incremental as it builds on active set methods.
The paper tackles the problem of sparse signal recovery by developing a primal-dual active set algorithm for nonconvex penalties, showing global convergence to the regression target under certain conditions and demonstrating superior efficiency and accuracy in numerical experiments compared to existing methods.
In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, including $\ell^0$, bridge, smoothly clipped absolute deviation, capped $\ell^1$ and minimax concavity penalty. First we establish the existence of a global minimizer for the related optimization problems. Then we derive a novel necessary optimality condition for the global minimizer using the associated thresholding operator. The solutions to the optimality system are coordinate-wise minimizers, and under minor conditions, they are also local minimizers. Upon introducing the dual variable, the active set can be determined using the primal and dual variables together. Further, this relation lends itself to an iterative algorithm of active set type which at each step involves first updating the primal variable only on the active set and then updating the dual variable explicitly. When combined with a continuation strategy on the regularization parameter, the primal dual active set method is shown to converge globally to the underlying regression target under certain regularity conditions. Extensive numerical experiments with both simulated and real data demonstrate its superior performance in efficiency and accuracy compared with the existing sparse recovery methods.