Two Proofs for Shallow Packings

Authors Kunal Dutta, Esther Ezra, Arijit Ghosh

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Kunal Dutta
Esther Ezra
Arijit Ghosh

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Kunal Dutta, Esther Ezra, and Arijit Ghosh. Two Proofs for Shallow Packings. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 96-110, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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 delta-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u, v in W is greater than delta, where delta > 0 is an integer parameter. The delta-packing number is then defined as the cardinality of the largest delta-separated subcollection of V. Haussler showed an asymptotically tight bound of Theta((n / delta)^d) on the delta-packing number if V has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, X' of 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(m^{d_1} k^{d-d_1}), 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 delta-packing number is O(n^{d_1} k^{d-d_1} / delta^d), 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.
  • Set systems of bounded primal shatter dimension
  • delta-packing & Haussler’s approach
  • relative approximations
  • Clarkson-Shor random sampling approach


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