Factorization Norms and Hereditary Discrepancy

TL;DR

This paper establishes a tight relationship between the $oldsymbol{ ext{γ}_2}$ norm and hereditary discrepancy, providing polynomial-time approximation algorithms and new bounds for discrepancy problems like the Tusnady problem in higher dimensions.

Contribution

It proves that the $ ext{γ}_2$ norm approximates hereditary discrepancy within logarithmic factors, introduces a polynomial-time approximation method, and improves lower bounds for the Tusnady problem in multiple dimensions.

Findings

$ ext{γ}_2(A) = O( ext{herdisc} extbf{ }A imes ext{log} m)$

$ ext{herdisc} extbf{ }A = O( ext{γ}_2(A) imes ext{sqrt}( ext{log} m))$

New lower bound of $oldsymbol{ ext{Ω}( ext{log}^{d-1} n)}$ for the $d$-dimensional Tusnady problem.

Abstract

The $\gamma_2$ norm of a real $m\times n$ matrix $A$ is the minimum number $t$ such that the column vectors of $A$ are contained in a $0$-centered ellipsoid $E\subseteq\mathbb{R}^m$ which in turn is contained in the hypercube $[-t, t]^m$. We prove that this classical quantity approximates the \emph{hereditary discrepancy} $\mathrm{herdisc}\ A$ as follows: $\gamma_2(A) = {O(\log m)}\cdot \mathrm{herdisc}\ A$ and $\mathrm{herdisc}\ A = O(\sqrt{\log m}\,)\cdot\gamma_2(A) $. Since $\gamma_2$ is polynomial-time computable, this gives a polynomial-time approximation algorithm for hereditary discrepancy. Both inequalities are shown to be asymptotically tight. We then demonstrate on several examples the power of the $\gamma_2$ norm as a tool for proving lower and upper bounds in discrepancy theory. Most notably, we prove a new lower bound of $\Omega(\log^{d-1} n)$ for the \emph{$d$-dimensional Tusn\'ady problem}, asking for the combinatorial discrepancy of an $n$-point set in $\mathbb{R}^d$ with respect to axis-parallel boxes. For $d>2$, this improves the previous best lower bound, which was of order approximately $\log^{(d-1)/2}n$, and it comes close to the best known upper bound of $O(\log^{d+1/2}n)$, for which we also obtain a new, very simple proof.