Extension of Correspondence Analysis to multiway data-sets through High Order SVD: a geometric framework

TL;DR

This paper extends Correspondence Analysis to multiway data using High Order SVD, providing a geometric interpretation and demonstrating its advantages over traditional matrix methods through real data examples.

Contribution

It introduces a geometric framework for tensor-based correspondence analysis using HOSVD, linking point clouds across tensor modes.

Findings

Point clouds in tensor modes are related through CA metrics.

Barycentric relations hold in the tensor framework.

Advantages and drawbacks are demonstrated with real data sets.

Abstract

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.