Non-interactive simulation of joint distributions: The Hirschfeld-Gebelein-Rényi maximal correlation and the hypercontractivity ribbon

Abstract

We consider the following problem: Alice and Bob observe sequences X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> and Y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> respectively where {(X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> )} <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> are drawn i.i.d. from P(x, y), and they output U and V respectively which is required to have a joint law that is close in total variation to a specified Q(u, v). One important technique to establish impossibility results for this problem is the Hirschfeld-Gebelein-Rényi maximal correlation which was considered by Witsen-hausen [1]. Hypercontractivity studied by Ahlswede and Gács [2] and reverse hypercontractivity recently studied by Mossel et al. [3] provide another approach for proving impossibility results. We consider the tightest impossibility results that can be obtained using hypercontractivity and reverse hypercontractivity and provide a necessary and sufficient condition on the source distribution P(x, y) for when this approach subsumes the maximal correlation approach. We show that the binary pair source distribution with symmetric noise satisfies this condition.