广州数学大讲坛第七期

第六十四讲——信阳师范大学余永超博士学术报告


题目：DCACO: an algorithm for designing incoherent redundant matrices

时间：2023年9月20日（星期三）晚上19:30-22:00

地点：腾讯会议（会议ID：257-689-662）

报告人：余永超 博士

摘要：The mutual coherence of a matrix, defined as the maximum absolute value of the normalized inner-products between different columns, is an important property that characterizes the similarity between different matrix columns. Redundant matrices with very low mutual coherence are referred to as incoherent redundant matrices which play an important role in mathematical signal processing tasks. The problem of minimizing the mutual coherence in a give matrix space where each matrix has normalized columns is called the coherence optimization problem. In this paper, we transform equivalently the coherence optimization problem as a rank constrained semidefinite ℓ∞-minimization problem. It is critical to analyze the projection operator onto the nonconvex set in the new matrix optimization constraints, i.e., the nonconvex set of symmetric positive semidefinite matrices whose rank is not greater than a give positive integer. By exploiting the projection operator, we express the nonconvex set mentioned above as the set of zero roots of a Difference of two Convex (DC) functions. For the convex function related to the projection operator in the DC function, we characterize its properties and obtain the concise form of its subdifferential. With the help of the DC function, a new algorithm based on DCA (DC Algorithms) is proposed to solve the Coherence Optimization problem, and thus the proposed algorithm is called DCACO for short. We also study the convergence analysis of DCACO. An advantage of DCACO is that subproblems in each iteration have closed-form solutions. Experimental results demonstrate that DCACO leads to state-of-art performance on generating highly incoherent redundant matrices, and DCACO can also compete with several other algorithms on designing optimized projection matrices for improving the performance of compressed sensing.

报告人简介：

余永超，信阳师范大学数学与统计学院讲师。2017 年毕业于西安交通大学数学与统计学院，获理学博士学位。2013 年和 2015 年分别在加拿大University of Alberta（阿尔伯塔大学）数学与统计学院和英国 University of Lincoln（林肯大学）计算机学院可计算智能实验室做访问学者。主要在凸分析与凸优化、数值代数、稀疏信号处理、机器学习等领域从事数学与信息科学的交叉研究。2019年获得国家自然科学基金青年项目。