学术报告报告题目: On Convergence of Iterative Thresholding Algorithms to Approximate Sparse Solution for Nonconvex Sparse Optimization
报 告人: 胡耀华教授
报告时间: 2023年 3月 12 日（星期日）9:30-10:15
报告地点：四号楼4318会议室
邀 请人: 潘少华
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数学学院
2023年3 月8日
报告摘要：Sparse optimization is a popular research topic in applied mathematics and optimization, and nonconvex sparse regularization problems have been extensively studied to ameliorate the statistical bias and enjoy robust sparsity promotion capability in vast applications. However, puzzled by the nonconvex and nonsmooth structure in nonconvex regularization problems, the convergence theory of their optimization algorithms is still far from completion: only the convergence to a stationary point was established in the literature, while there is still no theoretical evidence to guarantee the convergence to a global minimum or a true sparse solution.This talk aims to find an approximate true sparse solution of an under-determined 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 Lp penalty 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 L1 or L0 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.
报告人简介：胡耀华，现任深圳大学数学与统计学院教授，博士生导师，香港理工大学兼职博导，兼任中国运筹学会数学规划分会青年理事。主要从事连续优化理论、算法与应用研究，先后主持国家优秀青年科学基金、面上项目等4项，省市级科研项目多项。在SIAM Journal on Optimization, Journal of Machine Learning Research, Briefings in Bioinformatics等国际期刊发表论文40余篇，申请3项国家发明专利，开发多个生物信息学工具包与网页服务器。