## 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 ≤ d_{1}≤ d (this generalizes the standard notion of bounded primal shatter dimension when d_{1}= 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 d_{1}= 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 language | English |
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Pages (from-to) | 910-939 |

Number of pages | 30 |

Journal | Discrete and Computational Geometry |

Volume | 56 |

Issue number | 4 |

State | Published - 1 Dec 2016 |

Externally published | Yes |

## 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