联合分布非交互模拟的可判定性
Decidability of Non-Interactive Simulation of Joint Distributions
摘要：
我们提出了一个子类的“非交互式”的仿真问题，一类在信息论的问题，研究的可判定性结果。一个非交互式仿真问题由两个分布$P(x,y)$和$Q(u,v)$指定：目标是确定两个玩家Alice和Bob是否分别观察序列$X^n$和$Y^n$，其中$\{(X_i, Y_i)\}_{i=1}^n$被i.i.d.绘制。从$P(x,y)$可以分别生成具有联合分布的对$U$和$V$（彼此不通信），该联合分布在总变化上任意接近于$Q(u,v)$。即使当$P$和$Q$非常简单时：例如，$P$在三元组$\{(0,0), (0,1), (1,0)\}$上是均匀的，并且$Q$是“双对称二进制源”，即，$U$和$V$是具有相关性的统一$\pm 1$变量，例如$0.49$，如果$P$可以模拟$Q$，则是开放的。在这项工作中，我们表明，只要$P$是一个有限域上的分布和$Q$是一个$2 \times 2$分布，那么非交互式仿真问题是可判定的：具体地说，给定$δ> 0$，算法在由$P$和$δ$的某个函数限定的时间内运行，并且要么给出接近于$Q$的$δ$的非交互式仿真协议，要么断言没有协议得到$O(δ)$。靠近$Q$。对这样的结果的主要挑战是确定需要从$P(x,y)$中提取的样本的数量$n$上的显式（可计算的）收敛界限，以得到接近$Q$的$δ$。我们调用当代的布尔函数，如不变性原理和正则性引理的分析结果，以获得这样的明确的界限。
英文摘要：
We present decidability results for a sub-class of "non-interactive" simulation problems, a well-studied class of problems in information theory. A non-interactive simulation problem is specified by two distributions $P(x,y)$ and $Q(u,v)$: The goal is to determine if two players, Alice and Bob, that observe sequences $X^n$ and $Y^n$ respectively where $\{(X_i, Y_i)\}_{i=1}^n$ are drawn i.i.d. from $P(x,y)$ can generate pairs $U$ and $V$ respectively (without communicating with each other) with a joint distribution that is arbitrarily close in total variation to $Q(u,v)$. Even when $P$ and $Q$ are extremely simple: e.g., $P$ is uniform on the triples $\{(0,0), (0,1), (1,0)\}$ and $Q$ is a "doubly symmetric binary source", i.e., $U$ and $V$ are uniform $\pm 1$ variables with correlation say $0.49$, it is open if $P$ can simulate $Q$. In this work, we show that whenever $P$ is a distribution on a finite domain and $Q$ is a $2 \times 2$ distribution, then the non-interactive simulation problem is decidable: specifically, given $δ> 0$ the algorithm runs in time bounded by some function of $P$ and $δ$ and either gives a non-interactive simulation protocol that is $δ$-close to $Q$ or asserts that no protocol gets $O(δ)$-close to $Q$. The main challenge to such a result is determining explicit (computable) convergence bounds on the number $n$ of samples that need to be drawn from $P(x,y)$ to get $δ$-close to $Q$. We invoke contemporary results from the analysis of Boolean functions such as the invariance principle and a regularity lemma to obtain such explicit bounds.
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