TY - GEN

T1 - Two Proofs for Shallow Packings

AU - Dutta, Kunal

AU - Ezra, Esther

AU - Ghosh, Arijit

PY - 2015/6/1

Y1 - 2015/6/1

N2 - 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.

AB - 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.

KW - Clarkson-Shor random sampling approach

KW - Relative approximations

KW - Set systems of bounded primal shatter dimension

KW - δ-packing and Haussler s approach

UR - http://www.scopus.com/inward/record.url?scp=84958152725&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SOCG.2015.96

DO - 10.4230/LIPIcs.SOCG.2015.96

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AN - SCOPUS:84958152725

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 96

EP - 110

BT - 31st International Symposium on Computational Geometry, SoCG 2015

A2 - Pach, Janos

A2 - Pach, Janos

A2 - Arge, Lars

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 31st International Symposium on Computational Geometry, SoCG 2015

Y2 - 22 June 2015 through 25 June 2015

ER -