标题： 具有收敛性分析的稀疏信号恢复的截断Huber惩罚 显示英文标题

标题： Truncated Huber Penalty for Sparse Signal Recovery with Convergence Analysis

摘要： 从欠定系统中稀疏信号恢复在使用常规的L_0和L_1惩罚时面临重大挑战，主要由于计算复杂性和估计偏差。 本文引入了一种截断Huber惩罚，这是一种非凸度量，有效地弥合了无偏稀疏恢复与可微优化之间的差距。 所提出的惩罚对小幅值元素应用二次正则化，同时截断大幅值，避免在最优解处出现不可微点。 理论分析表明，对于适当选择的阈值，可通过常规惩罚恢复的任意s稀疏解在截断Huber函数下仍为局部最优解。 此性质允许为其他惩罚正则化函数开发的精确且鲁棒的恢复理论直接扩展到截断Huber函数。 为解决优化问题，我们开发了一种块坐标下降（BCD）算法，在火花条件下具有有限步收敛保证。 进行了数值实验以验证所提出方法的有效性和鲁棒性。 此外，我们将截断Huber惩罚模型扩展到梯度域，展示了其在信号去噪和图像平滑中的适用性。 显示英文摘要

摘要： Sparse signal recovery from under-determined systems presents significant challenges when using conventional L_0 and L_1 penalties, primarily due to computational complexity and estimation bias. This paper introduces a truncated Huber penalty, a non-convex metric that effectively bridges the gap between unbiased sparse recovery and differentiable optimization. The proposed penalty applies quadratic regularization to small entries while truncating large magnitudes, avoiding non-differentiable points at optima. Theoretical analysis demonstrates that, for an appropriately chosen threshold, any s-sparse solution recoverable via conventional penalties remains a local optimum under the truncated Huber function. This property allows the exact and robust recovery theories developed for other penalty regularization functions to be directly extended to the truncated Huber function. To solve the optimization problem, we develop a block coordinate descent (BCD) algorithm with finite-step convergence guarantees under spark conditions. Numerical experiments are conducted to validate the effectiveness and robustness of the proposed approach. Furthermore, we extend the truncated Huber-penalized model to the gradient domain, illustrating its applicability in signal denoising and image smoothing.