用于非 Lipschitz 组稀疏优化的快速迭代阈值化与支撑及尺度收缩算法（fits3）（I）：最小二乘保真情形
A fast iterative thresholding and support-and-scale shrinking algorithm (fits3) for non-lipschitz group sparse optimization (i): the case of least-squares fidelity
AI总结：
针对非凸非 Lipschitz 正则化的组稀疏最小二乘问题，提出无需解方程、具备收敛保证的 FITS3，并在保持恢复精度的同时显著降低大规模计算时间。
摘要：
我们考虑设计一种新的高效且易于实现的算法，以求解带有一类非凸、非 Lipschitz 正则项的一般组稀疏优化模型，该算法名为快速迭代阈值化与支撑及尺度收缩算法（FITS3）。本文聚焦于最小二乘保真的情形。FITS3 基于此类模型的下界理论，并结合阈值化操作、线性化和外推技术进行设计。FITS3 具有两个优点。首先，它非常高效，尤其适合大规模问题，因为它采用支撑及尺度收缩，且不需要求解任何线性或非线性系统。对于两个重要的特殊情形，FITS3 仅包含矩阵-向量乘法和软阈值化等简单计算。其次，在适当假设下，FITS3 算法具有序列收敛保证。数值实验以及与近期现有非 Lipschitz 组恢复算法的比较表明，所提出的 FITS3 达到了相近的恢复精度，但在中大规模问题上，其 CPU 时间仅约为第二快比较算法的一半。
英文摘要：
We consider to design a new efficient and easy-to-implement algorithm to solve a general group sparse optimization model with a class of non-convex non-Lipschitz regularizations, named as fast iterative thresholding and support-and-scale shrinking algorithm (FITS3). In this paper we focus on the case of a least-squares fidelity. FITS3 is designed from a lower bound theory of such models and by integrating thresholding operation, linearization and extrapolation techniques. The FITS3 has two advantages. Firstly, it is quite efficient and especially suitable for large-scale problems, because it adopts support-and-scale shrinking and does not need to solve any linear or nonlinear system. For two important special cases, the FITS3 contains only simple calculations like matrix-vector multiplication and soft thresholding. Secondly, the FITS3 algorithm has a sequence convergence guarantee under proper assumptions. The numerical experiments and comparisons to recent existing non-Lipschitz group recovery algorithms demonstrate that, the proposed FITS3 achieves similar recovery accuracies, but costs only around a half of the CPU time by the second fastest compared algorithm for median or large-scale problems.
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