Two Proofs for Shallow Packings

Kunal Dutta, Esther Ezra, Arijit Ghosh

Research output: Contribution to journalArticlepeer-review

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∈ W is greater than δ, where δ> 0 is an integer parameter. The δ-packing number is then defined as the cardinality of a largest δ-separated subcollection of V. Haussler showed an asymptotically tight bound of Θ ((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 ≤ k≤ m, the number of vectors of length at most k 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 “k-shallow” (all vector lengths are at most k), we show that its δ-packing number is O(nd1kd-d1/δd), matching Haussler’s bound for the special cases where d1= d or k= 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
Pages (from-to)910-939
Number of pages30
JournalDiscrete and Computational Geometry
Volume56
Issue number4
StatePublished - 1 Dec 2016
Externally publishedYes

Keywords

  • Clarkson–Shor property
  • Packing lemma and shallow packing lemma
  • Primal shatter function
  • Set systems of finite VC–dimension

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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