This paper presents an extension of Correspondence Analysis (CA) to tensors through High Order Singular Value Decomposition (HOSVD) from a geometric viewpoint. Correspondence analysis is a well-known tool, developed from principal component analysis, for studying contingency tables. Different algebraic extensions of CA to multi-way tables have been proposed over the years, nevertheless neglecting its geometric meaning. Relying on the Tucker model and the HOSVD, we propose a direct way to associate with each tensor mode a point cloud. We prove that the point clouds are related to each other. Specifically using the CA metrics we show that the barycentric relation is still true in the tensor framework. Finally two data sets are used to underline the advantages and the drawbacks of our strategy with respect to the classical matrix approaches.
翻译:本文件从几何角度将通信分析(CA)扩展至通过高顺序单值分解(HOSVD)的发光器,从高顺序单值分解(HOSVD)向发光器延伸至发光器。函光分析是从主要组成部分分析中开发的用于研究应急表的著名工具。多年来,CA向多向表提出了不同的代数扩展,但忽略了它的几何含义。根据塔克模型和HOSVD,我们建议一种直接的方式将每个发光模式的点云联系起来。我们证明点云是相互联系的。具体地说,我们使用CA指标显示,在高压框架中,赤心关系仍然真实存在。最后,使用两套数据来强调我们战略在经典矩阵方法方面的利弊。