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Kindler-Safra Theorem on the p-biased hypercube via agreement theorems Harsha, Prahladh
Description
Nisan and Szegedy showed that low degree Boolean functions are juntas, namely, they depend only on a constant number of their variables. Kindler and Safra showed a robust version of the above: low degree functions which are almost Boolean are close to juntas. We study the same question on the p-biased hypercube, for very small p. The p-biased hypercube is a product probability space in which each coordinate is 1 with probability p and 0 otherwise. In this space most of the measure is on n-bit strings whose Hamming weight about pn << n. It turns out that here new phenomena emerge. For example, the function x_1 + ... + x_n=p (where x_i \in {0,1}) is close to Boolean but not close to a junta. We characterize low degree functions that are almost Boolean and show that they are close to a new class of functions which we call sparse juntas. An interesting aspect of our proof is a new proof paradigm that relies on a local to global agreement theorem. We cover the p-biased hypercube by many smaller dimensional copies of the uniform hypercube, and approximate our function locally via the standard Kindler-Safra theorem for constant p. We then stitch the local approximations together into one global function that is a sparse junta. The stitching is made feasible via a new local-to-global agreement theorem, which is an extension of the classical direct product results to larger dimensions. Time permitting, I'll show another application of this paradigm: extending the classical AKKLR low-degree tests to the p-biased hypercube. Based on joint work with Irit Dinur and Yuval Filmus.
Item Metadata
Title |
Kindler-Safra Theorem on the p-biased hypercube via agreement theorems
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2018-08-13T15:02
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Description |
Nisan and Szegedy showed that low degree Boolean functions are juntas, namely, they depend only on a constant number of their variables. Kindler and Safra showed a robust version of the above: low degree functions which are almost Boolean are close to juntas.
We study the same question on the p-biased hypercube, for very small p. The p-biased hypercube is a product probability space in which each coordinate is 1 with probability p and 0 otherwise. In this space most of the measure is on n-bit strings whose Hamming weight about pn << n.
It turns out that here new phenomena emerge. For example, the function x_1 + ... + x_n=p (where x_i \in {0,1}) is close to Boolean but not close to a junta.
We characterize low degree functions that are almost Boolean and show that they are close to a new class of functions which we call sparse juntas.
An interesting aspect of our proof is a new proof paradigm that relies on a local to global agreement theorem. We cover the p-biased hypercube by many smaller dimensional copies of the uniform hypercube, and approximate our function locally via the standard Kindler-Safra theorem for constant p. We then stitch the local approximations together into one global function that is a sparse junta. The stitching is made feasible via a new local-to-global agreement theorem, which is an extension of the classical direct product results to larger dimensions.
Time permitting, I'll show another application of this paradigm: extending the classical AKKLR low-degree tests to the p-biased
hypercube.
Based on joint work with Irit Dinur and Yuval Filmus.
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Extent |
39.0
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Tata Institute of Fundamental Research, Mumbai
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Series | |
Date Available |
2019-03-28
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0377634
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International