On convergence of iterative thresholding algorithms to approximate sparse solution for composite nonconvex optimization
针对欠定线性系统，提出两种带延续和截断技术的迭代阈值算法，证明其收敛到近似真实稀疏解，并应用于SCAD、MCP等非凸正则化问题，数值实验显示效果优于标准近端梯度算法。
Abstract This paper aims to find an approximate true sparse solution of an underdetermined linear system. For this purpose, we propose two types of iterative thresholding algorithms with the continuation technique and the truncation technique respectively. We introduce a notion of limited shrinkage thresholding operator and apply it, together with the restricted isometry property, to show that the proposed algorithms converge to an approximate true sparse solution within a tolerance relevant to the noise level and the limited shrinkage magnitude. Applying the obtained results to nonconvex regularization problems with SCAD, MCP and $$\ell _p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> penalty ( $$0\le p \le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ) and utilizing the recovery bound theory, we establish the convergence of their proximal gradient algorithms to an approximate global solution of nonconvex regularization problems. The established results include the existing convergence theory for $$\ell _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> or $$\ell _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> regularization problems for finding a true sparse solution as special cases. Preliminary numerical results show that our proposed algorithms can find approximate true sparse solutions that are much better than stationary solutions that are found by using the standard proximal gradient algorithm.