基于傅里叶变换的非交互源模拟
On Non-Interactive Source Simulation via Fourier Transform
摘要：
考虑了非交互源模拟（NISS）方案。在这种情况下，一对分布式代理Alice和Bob观察基于联合分布$P_{X,Y}$生成的分布式二进制无记忆源$(X^d,Y^d)$。智能体希望产生一对离散随机变量$(U_d,V_d)$，其联合分布为$P_{U_d,V_d}$，使得当输入块长度$d$取为渐近大时，$P_{U_d,V_d}$在总变差距离上收敛到目标分布$Q_{U,V}$。在给定输入分布$P_{X,Y}$的情况下可以产生的分布$Q_{U,V}$的集合上获得内边界和外边界。为此，提供了从分布集$Q_{U,V}$到星凸集的并集的双射映射。通过利用证明技术从离散傅立叶分析以及一种新的随机舍入技术，内部和外部的边界推导出每个这些星凸集，并通过反转上述双射映射，$Q_{U,V}$和$P_{X,Y}$的必要和充分条件下，$Q_{U,V}$可以产生从$P_{X,Y}$。该界限适用于NISS方案，其中输出字母$\mathcal{U}$和$\mathcal{V}$具有任意有限大小。在二进制输出字母表的情况下，外边界恢复先前最知名的外边界。
英文摘要：
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.
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