  Nonsmooth sparsity constrained optimization encompasses a broad spectrum of
applications in machine learning. This problem is generally non-convex and
NP-hard. Existing solutions to this problem exhibit several notable
limitations, including their inability to address general nonsmooth problems,
tendency to yield weaker optimality conditions, and lack of comprehensive
convergence analysis. This paper considers Smoothing Proximal Gradient Methods
(SPGM) as solutions to nonsmooth sparsity constrained optimization problems.
Two specific variants of SPGM are explored: one based on Iterative Hard
Thresholding (SPGM-IHT) and the other on Block Coordinate Decomposition
(SPGM-BCD). It is shown that the SPGM-BCD algorithm finds stronger stationary
points compared to previous methods. Additionally, novel theories for analyzing
the convergence rates of both SPGM-IHT and SPGM-BCD algorithms are developed.
Our theoretical bounds, capitalizing on the intrinsic sparsity of the
optimization problem, are on par with the best-known error bounds available to
date. Finally, numerical experiments reveal that SPGM-IHT performs comparably
to current IHT-style methods, while SPGM-BCD consistently surpasses them.
  Keywords: Sparsity Recovery, Nonsmooth Optimization, Nonconvex Optimization,
Proximal Gradient Method, Block Coordinate Decomposition, Iterative Hard
Thresholding, Convergence Analysis
