讲座主题：Some Novel Results on Spectral Theory of New-Type Matrices and Applications

专家姓名：张立平

工作单位：清华大学

讲座时间：2025年11月15日 15:00-16:00

讲座地点：二教104

主办单位：烟台大学数学与信息科学学院

内容摘要：

In this talk, we mainly introduce some novel results on spectral theory of new-type matrix and application. Testing the positive definiteness of a multivariate form in automatic control is very hard problem. For solving such a problem, M-tensor and strong M-tensor were introduced. Based on the eigenvalue theory of M-tensors, Collatz-type method and smoothing Newton method were proposed, which has the rate of linear convergence and quadratic convergence, respectively. For efficiently handling color video recovery, quaternion tensor was introduced. A novel tensor product for third order quaternion tensors was proposed. With this tensor product, we further defined the singular value decomposition and the rank of a third order quaternion tensor. It was also proved that the best rank-k approximation of a third order quaternion tensor exists, which opens up a new way to analyze the structure of color video. Finally, some experiments are performed to demonstrate the low-rankness of color videos.

主讲人介绍：

张立平，清华大学长聘教授，博士生导师，研究方向优化理论算法及应用、机器学习、强化学习，在大规模高维数据处理建模、优化算法及张量特征值理论等方面取得了重要研究成果。已在JMLR, Math Program, Math Comput, SIAM J Matrix Anal Appl, SIAM J Optim, MOR等优化和计算数学国内外核心期刊发表论文60多篇，连续获得多项国家自然科学基金面上资助和1项科技部重点研发专项资助。曾获得教育部自然科学奖二等奖和北京市科学技术奖二等奖。