数值计算系列报告

报告题目： Provable Tensor-Train Format Tensor Completion by Riemannian Optimization

报告人：蔡剑锋 教授

报告时间：2022年5月31日（周二）15:00-16:00

报告地点：勤园21号楼304室    腾讯会议：642-279-048

报告摘要：The recent decade has witnessed the wide applications of Tensor-Train (TT) format tensors from diverse disciplines, among which tensor completion has drawn considerable attention. Numerous fast algorithms, including the Riemannian gradient descent (RGrad), have been proposed for the TT-format tensor completion. However, the theoretical guarantees of these algorithms are largely missing or sub-optimal, partly due to the complicated and recursive algebraic operations in TT-format decomposition. Moreover, existing results established for the tensors of other formats, for example, Tucker and CP, are inapplicable because the algorithms treating TT-format tensors are substantially different and more involved. In this talk, we provide theoretical guarantees of the convergence of RGrad algorithm for TT-format tensor completion, under a nearly optimal sample size condition. The RGrad algorithm converges linearly with a constant contraction rate that is free of tensor condition number without the necessity of re-conditioning. We also propose a novel approach, referred to as the sequential second-order moment method, to attain a warm initialization under a similar sample size requirement. Statistically (near) optimal rate is derived for RGrad algorithm if the observed entries consist of random sub-Gaussian noise. Numerical experiments confirm our theoretical discovery and showcase the computational speedup gained by the TT-format decomposition.

报告人简介：蔡剑锋，香港科技大学数学系教授。2000年获复旦大学学士学位，2007年获香港中文大学博士学位。曾先后在新加坡国立大学，美国洛杉矶加州大学，和美国爱荷华大学工作。2015年加入香港科技大学数学系。研究兴趣是数据科学和成像技术中的算法设计和分析。在2017年和2018年被评选为全球高被引学者。目前已经在国际顶级数学期刊J. Amer. Math. Soc.和国际著名期刊和会议如Appl. Comput. Hamon. Anal., SIAM J. Optim.., SIAM J. Imaging Sci., IEEE Trans. Image Process., IEEE Trans. Signal Process., CVPR, ICCV等发表100余篇论文。