A self-consistent-field iteration for MAXBET with an application to multi-view feature extraction
A self-consistent-field iteration for MAXBET with an application to multi-view feature extraction
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DOI：
10.1007/s10444-022-09929-3
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发表时间：
2022-03
影响因子：
1.7
通讯作者：
Xijun Ma;Chungen Shen;Li Wang;Lei-Hong Zhang;Ren-Cang Li
中科院分区：
文献类型：
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作者：
Xijun Ma;Chungen Shen;Li Wang;Lei-Hong Zhang;Ren-Cang Li
作为传统主成分分析的扩展，多视点典型相关分析的目标是对高维随机变量进行约简。文档类[12pt]{minimum}\usepackage{amsath}\usepackage{waysym}\usepackage{amsfonts}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setLong{oddsidemarin}{-69pt}\Begin{Document}$\黑体符号{S}_{i}\\ldots，M)$\end{Document}通过适当的投影矩阵，使得简化后的文档类[12pt]{minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$\boldsign{y}_{i}=X_{i}T}}\黑体符号{S}_{i}\在\mathbb{R}^{\ell}$\end{文档}中具有“最大相关性”。关于(i= ，2，…)相关性的各种度量，m)在MCCA中的应用。最早的准则之一是受xi，i= 1，2，…上的正交性约束的i和j之间的成对相关矩阵的所有迹之和由此产生的问题是在Stiefel流形的乘积上最大化齐次二次函数，称为MAXBET问题。首先将问题转化为具有特征向量依赖关系的耦合非线性特征值问题，然后用一种新的自洽场迭代方法进行求解。研究了SCF迭代的全局和局部收敛，并结合了标准本征值问题中成熟的计算技术，以产生更实际的实现。除了对各种合成问题进行了初步的数值评估外，SCF迭代在无监督学习的多视点特征提取中的应用也证明了该方法的有效性。
As an extension of the traditional principal component analysis, the multi-view canonical correlation analysis (MCCA) aims at reducingmhigh dimensional random variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\boldsymbol {s}_{i}\in \mathbb {R}^{n_{i}}~(i=1,2,\ldots ,m)$\end{document} by proper projection matricesso that themreduced ones \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\boldsymbol {y}_{i}=X_{i}^{\mathrm {T}}\boldsymbol {s}_{i}\in \mathbb {R}^{\ell }$\end{document} have the “maximal correlation.” Various measures of the correlation foryi(i= 1,2,…,m) in MCCA have been proposed. One of the earliest criteria is the sum of all traces of pair-wise correlation matrices betweenyiandyjsubject to the orthogonality constraints onXi,i= 1,2,…,m. The resulting problem is to maximize a homogeneous quadratic function over the product of Stiefel manifolds and is referred to asthe MAXBET problem. In this paper, the problem is first reformulated as a coupled nonlinear eigenvalue problem with eigenvector dependency (NEPv) and then solved by a novel self-consistent-field (SCF) iteration. Global and local convergences of the SCF iteration are studied and proven computational techniques in the standard eigenvalue problem are incorporated to yield more practical implementations. Besides the preliminary numerical evaluations on various types of synthetic problems, the efficiency of the SCF iteration is also demonstrated in an application to multi-view feature extraction for unsupervised learning.