TY - GEN

T1 - Concentration on the Boolean hypercube via pathwise stochastic analysis

AU - Eldan, Ronen

AU - Gross, Renan

N1 - Publisher Copyright:
© 2020 ACM.

PY - 2020/6/8

Y1 - 2020/6/8

N2 - We develop a new technique for proving concentration inequalities which relate between the variance and influences of Boolean functions. Using this technique, we first settle a conjecture of Talagrand, proving that g g gg-1,1g g¬ g?nghfg gxg g?dμ≥ C·g gfg g?·g glogg g g g g1g'i2g gf1/2, where hf(x) is the number of edges at x along which f changes its value, and i(f) is the influence of the i-th coordinate. Second, we strengthen several classical inequalities concerning the influences of a Boolean function, showing that near-maximizers must have large vertex boundaries. An inequality due to Talagrand states that for a Boolean function f, (f)≤ Cg'i=1ni(f)/1+log(1/i(f)). We give a lower bound for the size of the vertex boundary of functions saturating this inequality. As a corollary, we show that for sets that satisfy the edge-isoperimetric inequality or the Kahn-Kalai-Linial inequality up to a constant, a constant proportion of the mass is in the inner vertex boundary. Lastly, we improve a quantitative relation between influences and noise stability given by Keller and Kindler. Our proofs rely on techniques based on stochastic calculus, and bypass the use of hypercontractivity common to previous proofs.

AB - We develop a new technique for proving concentration inequalities which relate between the variance and influences of Boolean functions. Using this technique, we first settle a conjecture of Talagrand, proving that g g gg-1,1g g¬ g?nghfg gxg g?dμ≥ C·g gfg g?·g glogg g g g g1g'i2g gf1/2, where hf(x) is the number of edges at x along which f changes its value, and i(f) is the influence of the i-th coordinate. Second, we strengthen several classical inequalities concerning the influences of a Boolean function, showing that near-maximizers must have large vertex boundaries. An inequality due to Talagrand states that for a Boolean function f, (f)≤ Cg'i=1ni(f)/1+log(1/i(f)). We give a lower bound for the size of the vertex boundary of functions saturating this inequality. As a corollary, we show that for sets that satisfy the edge-isoperimetric inequality or the Kahn-Kalai-Linial inequality up to a constant, a constant proportion of the mass is in the inner vertex boundary. Lastly, we improve a quantitative relation between influences and noise stability given by Keller and Kindler. Our proofs rely on techniques based on stochastic calculus, and bypass the use of hypercontractivity common to previous proofs.

KW - Boolean analysis

KW - Concentration

KW - Isoperimetric inequality

KW - Pathwise analysis

UR - http://www.scopus.com/inward/record.url?scp=85086767479&partnerID=8YFLogxK

U2 - 10.1145/3357713.3384230

DO - 10.1145/3357713.3384230

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AN - SCOPUS:85086767479

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 208

EP - 221

BT - STOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

A2 - Makarychev, Konstantin

A2 - Makarychev, Yury

A2 - Tulsiani, Madhur

A2 - Kamath, Gautam

A2 - Chuzhoy, Julia

PB - Association for Computing Machinery

T2 - 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020

Y2 - 22 June 2020 through 26 June 2020

ER -