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

Published: 31 May 2009 Publication History

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.

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    cover image ACM Conferences
    STOC '09: Proceedings of the forty-first annual ACM symposium on Theory of computing
    May 2009
    750 pages
    ISBN:9781605585062
    DOI:10.1145/1536414
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    Published: 31 May 2009

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    1. epsilon-net
    2. explicit construction
    3. linear threshold function

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    May 31 - June 2, 2009
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    View all
    • (2015)Entropy of Weight Distributions of Small-Bias Spaces and PseudobinomialityComputing and Combinatorics10.1007/978-3-319-21398-9_39(495-506)Online publication date: 24-Jun-2015
    • (2011)Hardness results for agnostically learning low-degree polynomial threshold functionsProceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms10.5555/2133036.2133159(1590-1606)Online publication date: 23-Jan-2011
    • (2011)An FPTAS for #Knapsack and Related Counting ProblemsProceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science10.1109/FOCS.2011.32(817-826)Online publication date: 22-Oct-2011
    • (2010)Pseudorandom generators for polynomial threshold functionsProceedings of the forty-second ACM symposium on Theory of computing10.1145/1806689.1806749(427-436)Online publication date: 5-Jun-2010
    • (2010)Bounded Independence Fools Degree-2 Threshold FunctionsProceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science10.1109/FOCS.2010.8(11-20)Online publication date: 23-Oct-2010
    • (2009)Bounded Independence Fools Halfspaces2009 50th Annual IEEE Symposium on Foundations of Computer Science10.1109/FOCS.2009.68(171-180)Online publication date: Oct-2009
    • (2009)Improved Approximation of Linear Threshold FunctionsProceedings of the 2009 24th Annual IEEE Conference on Computational Complexity10.1109/CCC.2009.8(161-172)Online publication date: 15-Jul-2009
    • (2009)Pseudorandom Bit Generators That Fool Modular SumsProceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques10.1007/978-3-642-03685-9_46(615-630)Online publication date: 21-Aug-2009

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