Explicit construction of a small epsilon-net for linear threshold functions

Yuval Rabani; Amir Shpilka

doi:10.1145/1536414.1536502

Explicit construction of a small epsilon-net for linear threshold functions

Yuval Rabani^*, Amir Shpilka

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

14 Scopus citations

Abstract

We give explicit constructions of epsilon nets for linear threshold functions on the binary cube and on the unit sphere. The size of the constructed nets is polynomial in the dimension n and in 1/ε . To the best of our knowledge no such constructions were previously known. Our results match, up to the exponent of the polynomial, the bounds that are achieved by probabilistic arguments. As a corollary we also construct subsets of the binary cube that have size polynomial in n and covering radius of n/2 - c√n log n, for any constant c. This improves upon the well known construction of dual BCH codes that only guarantee covering radius of n/2 - c√n.

Original language	English
Title of host publication	STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing
Pages	649-658
Number of pages	10
DOIs	https://doi.org/10.1145/1536414.1536502
State	Published - 2009
Externally published	Yes
Event	41st Annual ACM Symposium on Theory of Computing, STOC '09 - Bethesda, MD, United States Duration: 31 May 2009 → 2 Jun 2009

Publication series

Name	Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)	0737-8017

Conference

Conference	41st Annual ACM Symposium on Theory of Computing, STOC '09
Country/Territory	United States
City	Bethesda, MD
Period	31/05/09 → 2/06/09

Keywords

Epsilon-net
Explicit construction
Linear threshold function

Access to Document

10.1145/1536414.1536502

Cite this

@inproceedings{e9210973fcae4893a2dc1786282de014,

title = "Explicit construction of a small epsilon-net for linear threshold functions",

abstract = "We give explicit constructions of epsilon nets for linear threshold functions on the binary cube and on the unit sphere. The size of the constructed nets is polynomial in the dimension n and in 1/ε . To the best of our knowledge no such constructions were previously known. Our results match, up to the exponent of the polynomial, the bounds that are achieved by probabilistic arguments. As a corollary we also construct subsets of the binary cube that have size polynomial in n and covering radius of n/2 - c√n log n, for any constant c. This improves upon the well known construction of dual BCH codes that only guarantee covering radius of n/2 - c√n.",

keywords = "Epsilon-net, Explicit construction, Linear threshold function",

author = "Yuval Rabani and Amir Shpilka",

year = "2009",

doi = "10.1145/1536414.1536502",

language = "אנגלית",

isbn = "9781605585062",

series = "Proceedings of the Annual ACM Symposium on Theory of Computing",

pages = "649--658",

booktitle = "STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing",

note = "41st Annual ACM Symposium on Theory of Computing, STOC '09 ; Conference date: 31-05-2009 Through 02-06-2009",

}

Rabani, Y & Shpilka, A 2009, Explicit construction of a small epsilon-net for linear threshold functions. in STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing. Proceedings of the Annual ACM Symposium on Theory of Computing, pp. 649-658, 41st Annual ACM Symposium on Theory of Computing, STOC '09, Bethesda, MD, United States, 31/05/09. https://doi.org/10.1145/1536414.1536502

Explicit construction of a small epsilon-net for linear threshold functions. / Rabani, Yuval; Shpilka, Amir.
STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing. 2009. p. 649-658 (Proceedings of the Annual ACM Symposium on Theory of Computing).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Explicit construction of a small epsilon-net for linear threshold functions

AU - Rabani, Yuval

AU - Shpilka, Amir

PY - 2009

Y1 - 2009

N2 - We give explicit constructions of epsilon nets for linear threshold functions on the binary cube and on the unit sphere. The size of the constructed nets is polynomial in the dimension n and in 1/ε . To the best of our knowledge no such constructions were previously known. Our results match, up to the exponent of the polynomial, the bounds that are achieved by probabilistic arguments. As a corollary we also construct subsets of the binary cube that have size polynomial in n and covering radius of n/2 - c√n log n, for any constant c. This improves upon the well known construction of dual BCH codes that only guarantee covering radius of n/2 - c√n.

AB - We give explicit constructions of epsilon nets for linear threshold functions on the binary cube and on the unit sphere. The size of the constructed nets is polynomial in the dimension n and in 1/ε . To the best of our knowledge no such constructions were previously known. Our results match, up to the exponent of the polynomial, the bounds that are achieved by probabilistic arguments. As a corollary we also construct subsets of the binary cube that have size polynomial in n and covering radius of n/2 - c√n log n, for any constant c. This improves upon the well known construction of dual BCH codes that only guarantee covering radius of n/2 - c√n.

KW - Epsilon-net

KW - Explicit construction

KW - Linear threshold function

UR - http://www.scopus.com/inward/record.url?scp=70350694417&partnerID=8YFLogxK

U2 - 10.1145/1536414.1536502

DO - 10.1145/1536414.1536502

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:70350694417

SN - 9781605585062

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 649

EP - 658

BT - STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing

T2 - 41st Annual ACM Symposium on Theory of Computing, STOC '09

Y2 - 31 May 2009 through 2 June 2009

ER -

Explicit construction of a small epsilon-net for linear threshold functions

Abstract

Publication series

Conference

Keywords

Access to Document

Other files and links

Fingerprint

Cite this