Statistics Papers

Document Type

Journal Article

Date of this Version

3-2010

Publication Source

Geometric and Functional Analysis

Volume

19

Issue

6

Start Page

1713

Last Page

1756

DOI

10.1007/s00039-010-0047-x

Abstract

In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recently obtained with O’Donnell and Oleszkiewicz for multilinear polynomials with low influences and bounded degree and on properties of multi-dimensional Gaussian distributions.

We present two applications of the new bounds to the theory of social choice. We show that Majority is asymptotically the most predictable function among all low influence functions given a random sample of the voters. Moreover, we derive an almost tight bound in the context of Condorcet aggregation and low influence voting schemes on a large number of candidates. In particular, we show that for every low influence aggregation function, the probability that Condorcet voting on k candidates will result in a unique candidate that is preferable to all others is k−1+o(1). This matches the asymptotic behavior of the majority function for which the probability is k−1−o(1).

A number of applications in hardness of approximation in theoretical computer science were obtained using the results derived here in subsequent work by Raghavendra and by Austrin and Mossel.

A different type of applications involves hyper-graphs and arithmetic relations in product spaces. For example, we show that if A ⊂ Znm is of low influences, then the number of k tuples (x1, . . . , xk) ∈ Ak satisfying ∑ki=1 xi ∈ Bn mod m where B ⊂ [m] satisfies |B| ≥ 2 is (1 ± o(1))P[A]k(mk−1|B|)n which is the same as if A were a random set of probability P[A]. Our results also show that for a general set A without any restriction on the influences there exists a set of coordinates S ⊂ [n] with |S| = O(1) such that if C = {x : ∃y ∈ A, y[n]\S} then the number of k-tuples (x1, . . ., xk) ∈ Ck satisfying ∑ki=1 xi ∈ Bn mod m is (1 ± o(1))P[C]k(mk−1|B|)n.

Copyright/Permission Statement

The final publication is available at Springer via http://dx.doi.org/10.1007/s00039-010-0047-x

Keywords

invariance, discrete harmonic analysis, voting, hardness of approximation, Gaussian, isoperimetric inequalities

Share

COinS
 

Date Posted: 27 November 2017

This document has been peer reviewed.