Relating multiway discrepancy and singular values of graphs and contingency tables

TL;DR

This paper establishes a bound linking the $k$-way discrepancy of a nonnegative array to the $k$th largest non-trivial singular value of its normalized form, extending previous results and applicable to graph adjacency matrices.

Contribution

It provides a new upper bound for singular values based on $k$-way discrepancy, generalizing prior work and including directed and weighted graphs.

Findings

Bound relates $k$-way discrepancy to singular values.

Extends results to directed and weighted graphs.

Improves the tightness of bounds for the $k=1$ case.

Abstract

The $k$-way discrepancy $\disc_k (\C)$ of a rectangular array $\C$ of nonnegative entries is the minimum of the maxima of the within- and between-cluster discrepancies that can be obtained by simultaneous $k$-clusterings (proper partitions) of its rows and columns. In the main theorem, irrespective of the size of $\C$, we give the following estimate for the $k$th largest non-trivial singular value of the normalized table: $s_k \le 9\disc_{k } (\C ) (k+2 -9k\ln \disc_{k } (\C ))$, provided $\disc_{k } (\C ) <1$ and $k\le \rk (\C )$. This statement is the converse of Theorem 7 of Bolla \cite{Bolla14}, and the proof uses some lemmas and ideas of Butler \cite{Butler}, where only the $k=1$ case is treated, in which case our upper bound is the tighter. The result naturally extends to the singular values of the normalized adjacency matrix of a weighted undirected or directed graph.