提出一种自适应软阈值算法的理论分析，解决LASSO问题无需调参。
Adaptive Iterative Soft-Thresholding Algorithm with the Median Absolute Deviation
- 用中位数绝对偏差估计噪声水平进行自适应阈值
- 证明算法具有局部线性收敛和全局收敛性
- 适用于信号处理与高维统计建模中的稀疏重构任务
自适应迭代软阈值算法（adaptive ISTA）是一种在无需显式调节正则化参数λ的情况下求解LASSO问题的常用方法。尽管该算法在实践中表现良好，但其理论研究较为匮乏。本文针对通过中位数绝对偏差（Median Absolute Deviation）估计噪声水平的阈值策略，对自适应ISTA进行了理论分析。我们揭示了算法不动点的性质，包括尺度等变性、非唯一性和局部稳定性；证明了局部线性收敛性，并探讨了其全局收敛行为。
原文摘要 · Abstract (English)
The adaptive Iterative Soft-Thresholding Algorithm (ISTA) has been a popular algorithm for finding a desirable solution to the LASSO problem without explicitly tuning the regularization parameter $λ$. Despite that the adaptive ISTA is a successful practical algorithm, few theoretical results exist. In this paper, we present the theoretical analysis on the adaptive ISTA with the thresholding strategy of estimating noise level by median absolute deviation. We show properties of the fixed points of the algorithm, including scale equivariance, non-uniqueness, and local stability, prove the local linear convergence guarantee, and show its global convergence behavior.
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