Two Proofs for Shallow Packings

Kunal Dutta, Esther Ezra, Arijit Ghosh

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

4 Scopus citations

Abstract

We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A δ-separated set of V is a subcollection W, s.t. The Hamming distance between each pair u, v 2 W is greater than δ, where δ > 0 is an integer parameter. The δ-packing number is then defined as the cardinality of the largest δ-separated subcollection of V. Haussler showed an asymptotically tight bound of φ((n/δ)d) on the δ-packing number if V has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, X′ ⊆ X of size m ≤ n and for any parameter 1 ≤ κ ≤ m, the number of vectors of length at most κ in the restriction of V to X′ is only O(md1kd-d1 ), for a fixed integer d > 0 and a real parameter 1 ≤ d1 ≤ d (this generalizes the standard notion of bounded primal shatter dimension when d1 = d). In this case when V is "κ-shallow" (all vector lengths are at most κ), we show that its δ-packing number is O(nd1kd-d1/δd), matching Haussler s bound for the special cases where d1 = d or κ = n. We present two proofs, the first is an extension of Haussler s approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler s proof.

Original languageEnglish
Title of host publication31st International Symposium on Computational Geometry, SoCG 2015
EditorsJanos Pach, Janos Pach, Lars Arge
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages96-110
Number of pages15
ISBN (Electronic)9783939897835
DOIs
StatePublished - 1 Jun 2015
Externally publishedYes
Event31st International Symposium on Computational Geometry, SoCG 2015 - Eindhoven, Netherlands
Duration: 22 Jun 201525 Jun 2015

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume34
ISSN (Print)1868-8969

Conference

Conference31st International Symposium on Computational Geometry, SoCG 2015
Country/TerritoryNetherlands
CityEindhoven
Period22/06/1525/06/15

Keywords

  • Clarkson-Shor random sampling approach
  • Relative approximations
  • Set systems of bounded primal shatter dimension
  • δ-packing and Haussler s approach

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