Non interactive simulation of correlated distributions is decidable

Abstract

A basic problem in information theory is the following: Let P = (X, Y) be an arbitrary distribution where the marginals X and Y are (potentially) correlated. Let Alice and Bob be two players where Alice gets samples {x i } i≥1 and Bob gets samples {y i } i≥1 and for all i, (x i , y i ) ∼ P. What joint distributions Q can be simulated by Alice and Bob without any interaction?Classical works in information theory by Gács-Körner and Wyner answer this question when at least one of P or Q is the distribution Eq (Eq is defined as uniform over the points (0, 0) and (1, 1)). However, other than this special case, the answer to this question is understood in very few cases. Recently, Ghazi, Kamath and Sudan showed that this problem is decidable for Q supported on {0, 1} × {0, 1}. We extend their result to Q supported on any finite alphabet. Moreover, we show that If Q can be simulated, our algorithm also provides a (non-interactive) simulation protocol.We rely on recent results in Gaussian geometry (by the authors) as well as a new smoothing argument inspired by the method of boosting from learning theory and potential function arguments from complexity theory and additive combinatorics.

Communication for Generating Correlation: A Unifying Survey

“…While the general problem of characterizing when a distribution P X1X2 can simulate Q U1U2 remains open, an algorithmic procedure for testing a "gap-version" of the problem when U 1 and U 2 are both binary has been proposed recently in [79]; it has been extended to the general case in [67]. At a high level, the procedure is to produce random variables with as large a maximal correlation as possible from P X1X2 while maintaining the marginals of the simulated distribution as close to Q U1 and Q U2 ; the algorithm either produces a sample from a distribution such that the variational distance with Q U1U2 is less than error parameter δ or claims that there is no procedure which can produce a sample with distribution within O(δ) of Q U1U2 .…”

Untitled

“…In the best of both worlds, we would be able to replace {0, 1} by [k] without adding the assumption that P is a Gaussian measure. Using the methods we develop here, this also turns out to be possible; this is done in a follow-up work [8].…”

“…We do not know how to prove the analogue of Theorem 2.2 in the case of negative correlations, although we do discuss some related results in the follow-up paper [8]. The reason for this failure is that a certain trick which works in the k = 2 setting fails for k ≥ 3: suppose instead of maximizing E[ f , P t f ], we attempt to minimize E[ f 1 , P t f 2 ] where f 1 and f 2 are not required to be the same function.…”

Three candidate plurality is stablest for small correlations

“…Though this statement might seem intuitively true, because many inequalities involving the Gaussian measure have low-dimensional optimisers, this statement has not been proven before. For example, Theorem 1.9 was listed as an open question in [DMN17,DMN18] and [GKR18]. Indeed, the lack of Theorem 1.9 has been one main obstruction to a solution of Conjectures 1.5 and 1.7.…”