报告题目：Implicit Regularization and Entrywise Convergence of Riemannian Optimization for Low Tucker-Rank Tensor Completion

主 讲 人：魏轲 复旦大学

报告时间：2021年12月22日（周三）10：00

报告地点：腾讯会议 ID：456 335 701

报告摘要：

This talk is concerned with the low Tucker-rank tensor completion problem, which is about reconstructing a tensor $\mathcal{T}\in\mathbb{R}^{n\times n\times n}$ of low multilinear rank from partially observed entries. We consider a manifold algorithm (i.e., Riemannian gradient method) for this problem and reveal an appealing implicit regularization phenomenon of non-convex optimization in low Tucker-rank tensor completion. More precisely, it has been rigorously proved that the iterates of the Riemannian gradient method stay in an incoherent region throughout all iterations provided the number of observed entries is essentially in the order of $O(n^{3/2})$. To the best of our knowledge, this is the first work that has shown the implicit regularization property of a non-convex method for low Tucker-rank completion under the nearly optimal sampling complexity. Additionally, the entrywise convergence of the method is further established. The analysis relies on the leave-one-out technique and the subspace projection structure within the algorithm. Some of technical results developed in the paper might be of broader interest in investigating the properties of other non-convex algorithms.

主讲人简介：

魏轲，复旦大学大数据学院青年研究员，博士生导师。2014年获得牛津大学博士学位，之后在香港科技大学（2014-2015）和加州大学戴维斯分校（2015-2017）从事博士后研究。主要研究兴趣为高维结构化数据处理算法与理论，多智能体强化学习算法与理论，数据科学的数理基础。其研究成果已发表在国际重要的应用数学和工程期刊上，包括 SIAM系列、IEEE系列、ACHA、MP、JMLR、IP等。先后获得了扬帆计划、自科青年基金、东方学者、中组部“青年拔尖人才计划”等项目的资助。

主办单位：山东大学金融研究院、数学学院

邀请人：吴盼玉

欢迎各位老师同学积极参加！