On Non-Interactive Source Simulation via Fourier Transform

TL;DR

This paper studies the non-interactive source simulation problem, providing bounds and conditions for when a target distribution can be generated from a given source using Fourier analysis and novel techniques.

Contribution

It introduces a Fourier-based framework with star-convex set mappings and a randomized rounding method to characterize achievable distributions in NISS.

Findings

Inner and outer bounds on distribution sets

Necessary and sufficient conditions for distribution simulation

Recovery of known bounds in binary cases

Abstract

The non-interactive source simulation (NISS) scenario is considered. In this scenario, a pair of distributed agents, Alice and Bob, observe a distributed binary memoryless source $(X^d,Y^d)$ generated based on joint distribution $P_{X,Y}$. The agents wish to produce a pair of discrete random variables $(U_d,V_d)$ with joint distribution $P_{U_d,V_d}$, such that $P_{U_d,V_d}$ converges in total variation distance to a target distribution $Q_{U,V}$ as the input blocklength $d$ is taken to be asymptotically large. Inner and outer bounds are obtained on the set of distributions $Q_{U,V}$ which can be produced given an input distribution $P_{X,Y}$. To this end, a bijective mapping from the set of distributions $Q_{U,V}$ to a union of star-convex sets is provided. By leveraging proof techniques from discrete Fourier analysis along with a novel randomized rounding technique, inner and outer bounds are derived for each of these star-convex sets, and by inverting the aforementioned bijective mapping, necessary and sufficient conditions on $Q_{U,V}$ and $P_{X,Y}$ are provided under which $Q_{U,V}$ can be produced from $P_{X,Y}$. The bounds are applicable in NISS scenarios where the output alphabets $\mathcal{U}$ and $\mathcal{V}$ have arbitrary finite size. In case of binary output alphabets, the outer-bound recovers the previously best-known outer-bound.