Size sensitive packing number for Hamming cube and its consequences
Abstract
We prove a size-sensitive version of Haussler's Packing lemma~\cite{Haussler92spherepacking} for set-systems with bounded primal shatter dimension, which have an additional {\em size-sensitive property}. This answers a question asked by Ezra~\cite{Ezra-sizesendisc-soda-14}. We also partially address another point raised by Ezra regarding overcounting of sets in her chaining procedure. As a consequence of these improvements, we get an improvement on the size-sensitive discrepancy bounds for set systems with the above property. Improved bounds on the discrepancy for these special set systems also imply an improvement in the sizes of {\em relative -approximations} and -samples.
Keywords
Cite
@article{arxiv.1412.3922,
title = {Size sensitive packing number for Hamming cube and its consequences},
author = {Kunal Dutta and Arijit Ghosh},
journal= {arXiv preprint arXiv:1412.3922},
year = {2014}
}
Comments
At the time of submission, we have become aware of a similar packing result proven simultaneously by Ezra. However, we note that our proof of the main packing lemma is quite different from hers. Also, the focus of our paper is on discrepancy bounds and sampling complexity