基于Eligible正则项的稀疏优化模型与算法及其应用  

Eligible regularized sparse optimization models,algorithms and application

作　　者：李倩 王国强 高雪瑞 白延琴[2] Qian Li;Guoqiang Wang;Xuerui Gao;Yanqin Bai

机构地区：[1]上海工程技术大学数理与统计学院,上海201620 [2]上海大学理学院数学系,上海200444

出　　处：《中国科学:数学》2025年第2期261-282,共22页Scientia Sinica:Mathematica

基　　金：国家自然科学基金(批准号:12171307和12201394)资助项目。

摘　　要：在当前机器学习、深度学习及统计学习的前沿研究中,众多核心科学难题能够借助基于正则项的稀疏优化模型进行有效表征.本文致力于探索这类基于正则项的稀疏优化模型、有效算法及实际应用场景,以期推动该领域发展.首先,创新性地提出Eligible正则项的概念.其次,建立基于可分和不可分Eligible正则项的精确恢复理论.接着,构建Eligible正则稀疏优化模型,并设计高效加速邻近梯度算法框架进行求解.最后,给出相位图分析和稀疏指数追踪实证分析的应用.At the forefront of current research encompassing machine learning,deep learning,and statistical learning,numerous pivotal scientific challenges can be effectively characterized through the employment of sparse optimization models grounded in regularization techniques.This paper commits to an in-depth exploration of such regularization-based sparse optimization models,their effective algorithms,and practical application scenarios,aspiring to propel advancements within this field.To this end,we first innovatively propose the Eligible regularization terms.Secondly,the accurate recovery theories based on separable and inseparable Eligible regularization terms are established.Subsequently,we construct the Eligible sparse regularization optimization models.Following this,we design the efficient proximal gradient algorithm and accelerated proximal gradient algorithm frameworks.Finally,empirical analysis results are conducted by phase diagram inspections and sparse index tracking.

关 键 词：稀疏优化 正则项 邻近梯度算法 加速邻近梯度算法 

分 类 号：O224[理学—运筹学与控制论]