An Upper Bound on ℓq Norms of Noisy Functions

Alex Samorodnitsky

doi:10.1109/TIT.2019.2944698

An Upper Bound on ℓq Norms of Noisy Functions

Alex Samorodnitsky^*

^*Corresponding author for this work

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

Let Tϵ, 0 ≤ ϵ ≤ 1/2, be the noise operator acting on functions on the boolean cube 0,1\n. Let f be a nonnegative function on 0,1n and let q ≥ 1. We upper bound the ℓ q norm of Tϵ f by the average ℓ q norm of conditional expectations of f, given sets of roughly 1-2 ϵ rq n variables, where r is an explicitly defined function of q. We describe some applications for error-correcting codes and for matroids. In particular, we derive an upper bound on the weight distribution of BEC-capacity achieving binary linear codes and their duals. This improves the known bounds on the linear-weight components of the weight distribution of constant rate binary Reed-Muller codes for all (constant) rates.

Original language	American English
Article number	8853269
Pages (from-to)	742-748
Number of pages	7
Journal	IEEE Transactions on Information Theory
Volume	66
Issue number	2
DOIs	https://doi.org/10.1109/TIT.2019.2944698
State	Published - Feb 2020

Bibliographical note

Funding Information:
Manuscript received November 19, 2018; accepted August 19, 2019. Date of publication September 30, 2019; date of current version January 20, 2020. This work was supported in part by ISF under Grant 1724/15. The author is with the School of Engineering and Computer Science, The Hebrew University of Jerusalem, Jerusalem 91904, Israel (e-mail: salex@cs.huji.ac.il). Communicated by E. Abbe, Associate Editor for Machine Learning. Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2019.2944698

Publisher Copyright:
© 1963-2012 IEEE.

Keywords

Binary codes
channel capacity

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abstract = "Let Tϵ, 0 ≤ ϵ ≤ 1/2, be the noise operator acting on functions on the boolean cube 0,1\n. Let f be a nonnegative function on 0,1n and let q ≥ 1. We upper bound the ℓ q norm of Tϵ f by the average ℓ q norm of conditional expectations of f, given sets of roughly 1-2 ϵ rq n variables, where r is an explicitly defined function of q. We describe some applications for error-correcting codes and for matroids. In particular, we derive an upper bound on the weight distribution of BEC-capacity achieving binary linear codes and their duals. This improves the known bounds on the linear-weight components of the weight distribution of constant rate binary Reed-Muller codes for all (constant) rates.",

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An Upper Bound on ℓq Norms of Noisy Functions

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