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 $P_{X^dY^d}$. 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 $O(\exp\operatorname{poly}(\frac{1}{\epsilon}))$. 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 $\Theta(\log{\frac{1}{\epsilon}})$, 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.

翻译：本文提出了一种用于非交互式源模拟问题的傅里叶分析框架。两个分布式智能体观测到根据联合分布$P_{X^dY^d}$抽取的一对序列$X^d$和$Y^d$。这些智能体旨在生成输出$U=f_d(X^d)$和$V=g_d(Y^d)$，其联合分布在总变差距离上充分接近目标分布$Q_{UV}$。现有研究表明，具有有限字母表输出的非交互式源模拟问题是可判定的。对于二进制输出的非交互式源模拟问题，已推导出输入复杂度的上界为$O(\exp\operatorname{poly}(\frac{1}{\epsilon}))$。本文在几类非交互式源模拟场景中研究了输入复杂度和算法设计问题。对于具有双对称二进制输入的二进制输出非交互式源模拟场景，证明了输入复杂度为$\Theta(\log{\frac{1}{\epsilon}})$，从而在输入复杂度上实现了超指数改进。文中给出了模拟函数对的显式刻画。针对一般有限输入场景，提出了一种能够显式找到模拟函数$(f_d(X^d),g_d(Y^d))$的构造性算法。该方法基于一种新颖的傅里叶分析框架。本文提供了多种独立同分布输入的非交互式源模拟场景的数值仿真结果。此外，为说明该傅里叶框架的广泛适用性，还给出了若干非独立同分布输入的实例，包括纠缠辅助的非交互式源模拟和马尔可夫输入的非交互式源模拟。