Convergence analysis for low rank partially orthogonal tensor approximation problem(低秩部分正交张量近似问题的收敛性分析)

时间:2021-10-13         阅读:

光华讲坛——社会名流与企业家论坛第5860期

主题:Convergence analysis for low rank partially orthogonal tensor approximation problem(低秩部分正交张量近似问题的收敛性分析)

主讲人:中国科学院数学与系统科学研究院 叶科副研究员

主持人:经济数学学院 车茂林副教授

时间:2021年10月19日(周二)09:00—10:00

直播平台及会议ID:腾讯会议:909679525;密码:1019

主办单位:经济数学学院 科研处

主讲人简介:

叶科,中国科学院数学与系统科学研究院副研究员,入选海外高层次人才引进计划(青年项目),中科院百人计划(C类),中科院基础研究领域青年团队计划,以及中科院“陈景润未来之星”。研究兴趣是代数几何及微分几何的在计算复杂度理论,(多重)线性代数,数值计算以及优化问题中的应用。工作主要发表于Adv. Math., FoCM, Math. Program., SIMAX, IEEE Info. Theory等重要国际期刊。

内容提要:

Low rank partially orthogonal tensor approximation (LRPOTA) is an important problem in tensor computations and their applications. It includes Low rank orthogonal tensor approximation (LROTA) problem as a special case. A classical and widely used algorithm for the LRPOTA problem is the alternating least square and polar decomposition method (ALS-APD). In this talk, we will introduce an improved version ALS-iAPD of the classical ALS-APD, for which all the following three fundamental properties will be addressed: (i) the algorithm converges globally and the whole sequence converges to a KKT point without any assumption; (ii) it exhibits an overall sublinear convergence with an explicit rate which is sharper than the usual O(1/k) for first order methods in optimization; (iii) more importantly, it converges R-linearly for a generic tensor without any assumption. I will explain how algebraic and differential geometric tools are used to obtain these results in optimization theory. This talk is based on joint works with Shenglong Hu.

低秩部分正交张量近似问题是张量计算及其应用中的一个重要问题。求解该类问题的经典算法是ALS+APD算法。在这个报告中,我们改进了ALS+APD算法,并将其记为ALS+iAPD算法。ALS+iAPD算法有如下性质:1、算法是全局收敛到KKT点;2、这个算法是次线性收敛的;3、对于大多数张量来说,这个算法是R线性收敛的。