Adaptive Iterative Hard Thresholding for Least Absolute Deviation Problems with Sparsity Constraints

Constrained Least absolute deviation (LAD) problems often arise from sparse regression of statistical prediction and compressed sensing literature. It is challenging to solve LAD problems with sparsity constraints directly due to non-smoothness of objective functions and non-convex feasible sets. We provide an adaptive iterative hard thresholding (AIHT1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{AIHT}\,}}_1$$\end{document}) method to solve LAD problems with sparsity constraints. The sequence generated by AIHT1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{AIHT}\,}}_1$$\end{document} converges to ground truth linearly under the l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l_1$$\end{document} restricted isometry property condition. Then we apply our analysis method to the binary iterative hard thresholding (BIHT) algorithm in one-bit compressed sensing. We obtain a tighter error bound compared with our previous work on BIHT. To some extent, our results can explain the efficiency of BIHT in recovering sparse vectors and make up for the deficiency of the theoretical guarantee of BIHT. Finally, numerical examples demonstrate the validity of our convergence analysis.