On Non-Interactive Simulation of Distributed Sources With Finite Alphabets

This work presents a Fourier analysis framework for the non-interactive source simulation (NISS) problem. Two distributed agents observe a pair of sequences $X^{d}$ and $Y^{d}$ drawn according to a joint distribution $P_{X^{d}Y^{d}}$ . The agents aim to generate outputs $U=f_{d}(X^{d})$ and $V=g_{d}(Y^{d})$ with a joint distribution sufficiently close in total variation to a target distribution $Q_{UV}$ . Existing works have shown that the NISS problem with finite-alphabet outputs is decidable. For the binary-output NISS, an upper-bound to the input complexity was derived which is $O\left ({{\exp \mathrm {poly}\left ({{\frac {1}{\epsilon }}}\right)}}\right)$ . In this work, the input complexity and algorithm design are addressed in several classes of NISS scenarios. For binary-output NISS scenarios with doubly-symmetric binary inputs, it is shown that the input complexity is $\Theta \left ({{\log {\frac {1}{\epsilon }}}}\right)$ , thus providing a super-exponential improvement in input complexity. An explicit characterization of the simulating pair of functions is provided. For general finite-input scenarios, a constructive algorithm is introduced that explicitly finds the simulating functions $(f_{d}(X^{d}),g_{d}(Y^{d}))$ . The approach relies on a novel Fourier analysis framework. Various numerical simulations of NISS scenarios with IID inputs are provided. Furthermore, to illustrate the general applicability of the Fourier framework, several examples with non-IID inputs, including entanglement-assisted NISS and NISS with Markovian inputs are provided.