Factorization Norms and Hereditary Discrepancy

@article{Matousek2014FactorizationNA,
  title={Factorization Norms and Hereditary Discrepancy},
  author={Jir{\'i} Matousek and Aleksandar Nikolov and Kunal Talwar},
  journal={ArXiv},
  year={2014},
  volume={abs/1408.1376}
}
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… CONTINUE READING
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