Invariance principle on the slice

Yuval Filmus; Guy Kindler; Elchanan Mossel; Karl Wimmer

doi:10.1145/3186590

Invariance principle on the slice

Yuval Filmus, Guy Kindler, Elchanan Mossel, Karl Wimmer

The Rachel and Selim Benin School of Engineering and Computer Science

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

8 Scopus citations

Abstract

The non-linear invariance principle of Mossel, O'Donnell, and Oleszkiewicz establishes that if f (x₁, . . . , x_n) is a multilinear low-degree polynomial with low influences, then the distribution of f (B₁, . . . , B_n ) is close (in various senses) to the distribution of f (G₁, . . . , G_n ), where B_i ∈_R{-1, 1} are independent Bernoulli random variables and G_i ~ N(0, 1) are independent standard Gaussians. The invariance principle has seen many applications in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans-Williamson algorithm for MAX-CUT is optimal under the Unique Games Conjecture. More generally, MOO's invariance principle works for any two vectors of hypercontractive random variables (x₁, . . . , x_n ), (Y₁, . . . ,Y_n ) such that (i) Matching moments: X_i and Y_i have matching first and second moments and (ii) Independence: the variablesx₁, . . . , x_n are independent, as are Y₁, . . . ,Y_n. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions (x₁, . . . , x_n ) in which the individual coordinates are not independent. A common example is the uniform distribution on the slice(^[n] _k which consists of all vectors (x₁, . . . , x_n ) ∈ {0, 1}ⁿ with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdös-Ko-Rado theorems), and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X₁, . . . ,X_n ) is the uniform distribution on a slice(^[n] _pn) and (Y₁, . . . ,Y_n ) consists either of n independent Ber(p) random variables, or of n independent N(p,p(1 - p)) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain's tail theorem, a version of the Kindler-Safra structural theorem, and a stability version of the t-intersecting Erdös-Ko-Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra, and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.

Original language	American English
Article number	11
Journal	ACM Transactions on Computation Theory
Volume	10
Issue number	3
DOIs	https://doi.org/10.1145/3186590
State	Published - Jun 2018

Bibliographical note

Funding Information:
This material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. E. M. would like to acknowledge the support of the following grants: NSF grants DMS 1106999 and CCF 1320105, DOD ONR grant N00014-14-1-0823, and grant 328025 from the Simons Foundation. K.W. would like to acknowledge the support of NSF grant CCF 1117079.

Funding Information:
Y. F. would like to mention that this material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. E. M. would like to acknowledge the support of the following grants: NSF grants DMS 1106999 and CCF 1320105, DOD ONR grant N00014-14-1-0823, and grant 328025 from the Simons Foundation. K. W. would like to acknowledge the support of NSF grant CCF 1117079. Authors’ addresses: Y. Filmus, Technion – Israel Institute of Technology, Computer Science Department, Haifa, Israel; email: yuvalfi@cs.technion.ac.il; G. Kindler, The Hebrew University, School of Computer Science and Engineering, Jerusalem, Israel; email: gkindler@cs.huji.ac.il; E. Mossel, Massachusetts Institute of Technology, Mathematics, Institute for Data, Systems, and Society, Cambridge, Massachusetts; email: elmos@mit.edu; K. Wimmer, Duquesne University, Department of Mathematics and Computer Science, Pittsburgh, Pennsylvania; email: wimmerk@duq.edu. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2018 ACM 1942-3454/2018/04-ART11 $15.00 https://doi.org/10.1145/3186590

Publisher Copyright:
© 2018 ACM.

Keywords

Analysis of Boolean functions
Invariance principle
Johnson scheme
Slice

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10.1145/3186590

Cite this

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title = "Invariance principle on the slice",

abstract = "The non-linear invariance principle of Mossel, O'Donnell, and Oleszkiewicz establishes that if f (x1, . . . , xn) is a multilinear low-degree polynomial with low influences, then the distribution of f (B1, . . . , Bn ) is close (in various senses) to the distribution of f (G1, . . . , Gn ), where Bi ∈R{-1, 1} are independent Bernoulli random variables and Gi ~ N(0, 1) are independent standard Gaussians. The invariance principle has seen many applications in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans-Williamson algorithm for MAX-CUT is optimal under the Unique Games Conjecture. More generally, MOO's invariance principle works for any two vectors of hypercontractive random variables (x1, . . . , xn ), (Y1, . . . ,Yn ) such that (i) Matching moments: Xi and Yi have matching first and second moments and (ii) Independence: the variablesx1, . . . , xn are independent, as are Y1, . . . ,Yn. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions (x1, . . . , xn ) in which the individual coordinates are not independent. A common example is the uniform distribution on the slice([n] k which consists of all vectors (x1, . . . , xn ) ∈ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erd{\"o}s-Ko-Rado theorems), and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X1, . . . ,Xn ) is the uniform distribution on a slice([n] pn) and (Y1, . . . ,Yn ) consists either of n independent Ber(p) random variables, or of n independent N(p,p(1 - p)) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain's tail theorem, a version of the Kindler-Safra structural theorem, and a stability version of the t-intersecting Erd{\"o}s-Ko-Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra, and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.",

keywords = "Analysis of Boolean functions, Invariance principle, Johnson scheme, Slice",

author = "Yuval Filmus and Guy Kindler and Elchanan Mossel and Karl Wimmer",

note = "Funding Information: This material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. E. M. would like to acknowledge the support of the following grants: NSF grants DMS 1106999 and CCF 1320105, DOD ONR grant N00014-14-1-0823, and grant 328025 from the Simons Foundation. K.W. would like to acknowledge the support of NSF grant CCF 1117079. Funding Information: Y. F. would like to mention that this material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. E. M. would like to acknowledge the support of the following grants: NSF grants DMS 1106999 and CCF 1320105, DOD ONR grant N00014-14-1-0823, and grant 328025 from the Simons Foundation. K. W. would like to acknowledge the support of NSF grant CCF 1117079. Authors{\textquoteright} addresses: Y. Filmus, Technion – Israel Institute of Technology, Computer Science Department, Haifa, Israel; email: yuvalfi@cs.technion.ac.il; G. Kindler, The Hebrew University, School of Computer Science and Engineering, Jerusalem, Israel; email: gkindler@cs.huji.ac.il; E. Mossel, Massachusetts Institute of Technology, Mathematics, Institute for Data, Systems, and Society, Cambridge, Massachusetts; email: elmos@mit.edu; K. Wimmer, Duquesne University, Department of Mathematics and Computer Science, Pittsburgh, Pennsylvania; email: wimmerk@duq.edu. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. {\textcopyright} 2018 ACM 1942-3454/2018/04-ART11 $15.00 https://doi.org/10.1145/3186590 Publisher Copyright: {\textcopyright} 2018 ACM.",

year = "2018",

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doi = "10.1145/3186590",

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N1 - Funding Information: This material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. E. M. would like to acknowledge the support of the following grants: NSF grants DMS 1106999 and CCF 1320105, DOD ONR grant N00014-14-1-0823, and grant 328025 from the Simons Foundation. K.W. would like to acknowledge the support of NSF grant CCF 1117079. Funding Information: Y. F. would like to mention that this material is based upon work supported by the National Science Foundation under agreement No. DMS-1128155. E. M. would like to acknowledge the support of the following grants: NSF grants DMS 1106999 and CCF 1320105, DOD ONR grant N00014-14-1-0823, and grant 328025 from the Simons Foundation. K. W. would like to acknowledge the support of NSF grant CCF 1117079. Authors’ addresses: Y. Filmus, Technion – Israel Institute of Technology, Computer Science Department, Haifa, Israel; email: yuvalfi@cs.technion.ac.il; G. Kindler, The Hebrew University, School of Computer Science and Engineering, Jerusalem, Israel; email: gkindler@cs.huji.ac.il; E. Mossel, Massachusetts Institute of Technology, Mathematics, Institute for Data, Systems, and Society, Cambridge, Massachusetts; email: elmos@mit.edu; K. Wimmer, Duquesne University, Department of Mathematics and Computer Science, Pittsburgh, Pennsylvania; email: wimmerk@duq.edu. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. © 2018 ACM 1942-3454/2018/04-ART11 $15.00 https://doi.org/10.1145/3186590 Publisher Copyright: © 2018 ACM.

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N2 - The non-linear invariance principle of Mossel, O'Donnell, and Oleszkiewicz establishes that if f (x1, . . . , xn) is a multilinear low-degree polynomial with low influences, then the distribution of f (B1, . . . , Bn ) is close (in various senses) to the distribution of f (G1, . . . , Gn ), where Bi ∈R{-1, 1} are independent Bernoulli random variables and Gi ~ N(0, 1) are independent standard Gaussians. The invariance principle has seen many applications in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans-Williamson algorithm for MAX-CUT is optimal under the Unique Games Conjecture. More generally, MOO's invariance principle works for any two vectors of hypercontractive random variables (x1, . . . , xn ), (Y1, . . . ,Yn ) such that (i) Matching moments: Xi and Yi have matching first and second moments and (ii) Independence: the variablesx1, . . . , xn are independent, as are Y1, . . . ,Yn. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions (x1, . . . , xn ) in which the individual coordinates are not independent. A common example is the uniform distribution on the slice([n] k which consists of all vectors (x1, . . . , xn ) ∈ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdös-Ko-Rado theorems), and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X1, . . . ,Xn ) is the uniform distribution on a slice([n] pn) and (Y1, . . . ,Yn ) consists either of n independent Ber(p) random variables, or of n independent N(p,p(1 - p)) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain's tail theorem, a version of the Kindler-Safra structural theorem, and a stability version of the t-intersecting Erdös-Ko-Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra, and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.

AB - The non-linear invariance principle of Mossel, O'Donnell, and Oleszkiewicz establishes that if f (x1, . . . , xn) is a multilinear low-degree polynomial with low influences, then the distribution of f (B1, . . . , Bn ) is close (in various senses) to the distribution of f (G1, . . . , Gn ), where Bi ∈R{-1, 1} are independent Bernoulli random variables and Gi ~ N(0, 1) are independent standard Gaussians. The invariance principle has seen many applications in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans-Williamson algorithm for MAX-CUT is optimal under the Unique Games Conjecture. More generally, MOO's invariance principle works for any two vectors of hypercontractive random variables (x1, . . . , xn ), (Y1, . . . ,Yn ) such that (i) Matching moments: Xi and Yi have matching first and second moments and (ii) Independence: the variablesx1, . . . , xn are independent, as are Y1, . . . ,Yn. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions (x1, . . . , xn ) in which the individual coordinates are not independent. A common example is the uniform distribution on the slice([n] k which consists of all vectors (x1, . . . , xn ) ∈ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdös-Ko-Rado theorems), and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X1, . . . ,Xn ) is the uniform distribution on a slice([n] pn) and (Y1, . . . ,Yn ) consists either of n independent Ber(p) random variables, or of n independent N(p,p(1 - p)) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain's tail theorem, a version of the Kindler-Safra structural theorem, and a stability version of the t-intersecting Erdös-Ko-Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra, and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.

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Invariance principle on the slice

Abstract

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