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)

時間:2021-10-13         阅读:

光華講壇——社會名流與企業家論壇第5860期

主題: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)

主講人:中国科学院数学与系统科學研究院 叶科副研究员

主持人:经济数学学院 车茂林副教授

時間:2021年10月19日(周二)09:00—10:00

直播平台及會議ID:騰訊會議:909679525;密碼:1019

主辦單位:经济数学学院 科研处

主講人簡介:

叶科,中国科学院数学与系统科學研究院副研究员,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。

內容提要:

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.

低秩部分正交張量近似問題是張量計算及其應用中的一個重要問題。求解該類問題的經典算法是ALS+APD算法。在這個報告中,我們改進了ALS+APD算法,並將其記爲ALS+iAPD算法。ALS+iAPD算法有如下性質:1、算法是全局收斂到KKT點;2、這個算法是次線性收斂的;3、對于大多數張量來說,這個算法是R線性收斂的。