摘要
本文概述信息压缩背景下从一阶到高阶主成分分析的统计原理,并从3个不同角度揭示各阶主成分分析的特点和局限,同时指出可能的研究方向.首先以相似的结构综述一阶到高阶主成分分析的原理及已有的发展,并进一步分析其内在相似统计意义,揭示其共性结构—"勾股定理",并以此为基础展示主成分分析的两种等价表达—"变异性最大"与"平方损失最小".其次深入分析了一阶到高阶PCA (principle component analysis)的可能发展:从"勾股定理"出发, PCA可以推广到更一般损失函数形式的"稳健"或"稀疏"类PCA;从张量分解与主成分分析之间的关联出发, PCA可以为构造不同的张量分解提供新思路;从分析一阶到高阶主成分分析在揭示"各向异性"结构上的固有局限出发, PCA能够推广到更有价值的"深度"
In this paper,a statistical technique of principal component analysis (PCA)from one-order to higherorder data under the background of information compression is summarized,its characteristics and limitations of each order PCA are revealed from three different perspectives,and some possible research directions are pointed out.Firstly,the technique and some existing developments are summarized by a similar structure,their intrinsic similar statistical significance is further analyzed,and the common structure--the Pythagorean theorem that shows two equivalent expressions of PCA--"maximizing variability"and "minimizing SClUare loss"is shown. Secondly,this paper analyzes three important angles of PCA:the first one starts from the perspective of the Pythagorean theorem and further points out that the PCA can be extended to more general loss function-- "robust"or "sparse"PCA;the second view reveals the relationship between tensor decomposition and PCA, which leads to a new idea of constructing tensor decomposition from the PCA perspective;the last view shows that higher-order PCA has a limitation to reveal anisotropic structure and further points out that a new method, "depth PCA,"can be used to conquer this limitation.
作者
夏志明
徐宗本
Zhiming XIA;Zongben XU(School of Mathematics and Statistics,Xi'an Jiaotong University,Xi'an 710049,China;School of Mathematics,Northwest University,Xi'an 710027,China)
出处
《中国科学：信息科学》
CSCD
北大核心
2018年第12期1622-1633,共12页
Scientia Sinica(Informationis)
基金
国家自然科学基金(批准号:11771353
11201372
91330204
11690011
11626252)资助项目
关键词
主成分分析
信息压缩
高阶张量
Tucker分解
各向异性
principle component analysis
information compression
high-order tensor
Tucker decomposition
anisotropy
作者简介
Zhiming XIA was born in 1978.He obtained his Ph.D.degree in statistics from Northwest University,Xi'an,in 2009.He currently serves as a professor of statistics at Northwest University.His research interests include tensor data processing,change-point analysis,distributed data processing,statistical process control,and nonparametric statistics;通信作者:徐宗本,Zongben XU was born in 1955.He obtained his Ph.D.degree in mathematics from Xi'an Jiaotong University,Xi'an, in 1987.He currently serves as a professor of mathematics & computer science and a Ph.D.supervisor,chief scientist of National Basic Research Program of China,and director of the Institute for Information and System Sciences at Xi'an Jiaotong University,Xi'an.He is an academician of the Chinese Academyof Sciences and a member of the New York Academy of Sciences. His current research interests include nonlinear functional analysis,mathematical foundations of information technology,and computational intelligence.E-mail:statxzm@nwu.edu.cn.
