On Non-Interactive Source Simulation via Fourier Transform

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 ${\mathcal{U}}{\text{and}}{\mathcal{V}}$ have arbitrary finite size. In case of binary output alphabets, the outer-bound recovers the previously best-known outer-bound.