On Non-Interactive Source Simulation via Fourier Transform
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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 𝒰 and 𝒱 have arbitrary finite size. In case of binary output alphabets, the outer-bound recovers the previously best-known outer-bound.
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