TY - GEN
T1 - Invariance principle on the slice
AU - Filmus, Yuval
AU - Kindler, Guy
AU - Mossel, Elchanan
AU - Wimmer, Karl
N1 - Publisher Copyright:
© Yuval Filmus, Guy Kindler, Elchanan Mossel, and Karl Wimmer.
PY - 2016/5/1
Y1 - 2016/5/1
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 ϵR {-1, 1} are independent Bernoulli random variables and Gi ∼N(0, 1) are independent standard Gaussians. The invariance principle has seen many application in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans-Williamson algorithm for MAXCUT 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, (ii) Independence: The variables X1, . . . ,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) ϵ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdos-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 Erdos-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 ϵR {-1, 1} are independent Bernoulli random variables and Gi ∼N(0, 1) are independent standard Gaussians. The invariance principle has seen many application in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans-Williamson algorithm for MAXCUT 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, (ii) Independence: The variables X1, . . . ,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) ϵ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdos-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 Erdos-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.
KW - Analysis of boolean functions
KW - Invariance principle
KW - Johnson association scheme
KW - The slice
UR - http://www.scopus.com/inward/record.url?scp=84973352067&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.CCC.2016.15
DO - 10.4230/LIPIcs.CCC.2016.15
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AN - SCOPUS:84973352067
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 15:1-15:10
BT - 31st Conference on Computational Complexity, CCC 2016
A2 - Raz, Ran
T2 - 31st Conference on Computational Complexity, CCC 2016
Y2 - 29 May 2016 through 1 June 2016
ER -