SVD, discrepancy, and regular structure of contingency tables

TL;DR

This paper links the SVD of normalized contingency tables to biclustering, providing bounds on discrepancy and regularity, extending previous results to multi-cluster and symmetric cases.

Contribution

It introduces a main theorem connecting volume-regularity constants to SVD, extending discrepancy estimation to multi-cluster contingency tables and symmetric graph cases.

Findings

The SVD of normalized tables bounds discrepancy in biclustering.

Extension of Butler's result to multi-cluster contingency tables.

Application to symmetric graph structures with eigenvalue analysis.

Abstract

We will use the factors obtained by correspondence analysis to find biclustering of a contingency table such that the row-column cluster pairs are regular, i.e., they have small discrepancy. In our main theorem, the constant of the so-called volume-regularity is related to the SVD of the normalized contingency table. Our result is applicable to two-way cuts when both the rows and columns are divided into the same number of clusters, thus extending partly the result of Butler estimating the discrepancy of a contingency table by the second largest singular value of the normalized table (one-cluster, rectangular case), and partly a former result of the author for estimating the constant of volume-regularity by the structural eigenvalues and the distances of the corresponding eigen-subspaces of the normalized modularity matrix of an edge-weighted graph (several clusters, symmetric case).