组合伯努利工厂
Combinatorial Bernoulli Factories
AI总结：
本文研究了多面体的伯努利工厂问题，证明其存在当且仅当多面体是单位立方体与仿射子空间的交集，并利用 Bernstein 多项式与 zonotope 平铺几何的联系，为完美匹配多面体构造了显式工厂。
摘要：
伯努利工厂是一种算法过程，用于在仅有伯努利访问参数的条件下对某些随机变量进行精确采样。对参数 $p \in [0,1]$ 的伯努利访问意味着算法不知道 $p$，但可以对均值为 $p$ 的伯努利随机变量的独立抽取进行采样访问。在本文中，我们研究了多面体的伯努利工厂问题：给定对某多面体 $P\subset [0,1]^n$ 中向量 $x\in P$ 的伯努利访问，输出一个随机化顶点，使得第 $i$ 个坐标的期望值\emph{精确}等于 $x_i$。例如，对于完美匹配多面体这一特例，给定对双随机矩阵 $[x_{ij}]$ 各元素的伯努利访问，要求采样一个匹配，使得边 $(i,j)$ 出现在匹配中的概率精确等于 $x_{ij}$。我们证明，多面体 $P$ 存在伯努利工厂当且仅当 $P$ 是 $[0,1]^n$ 与仿射子空间的交集。我们的构造基于该问题的代数表述，涉及识别一族 Bernstein 多项式（每个顶点一个），这些多项式在 $P$ 上满足特定的代数恒等式。我们构造背后的主要技术工具是这些多项式与 zonotope 平铺几何之间的联系。我们应用这些结果为完美匹配多面体构造了一个显式工厂。所得工厂与树形图的组合枚举有深刻联系，可能具有独立的研究价值。对于 $k$-均匀拟阵多面体，我们恢复了一种在统计学中称为 Sampford sampling 的采样过程。
英文摘要：
A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter $p \in [0,1]$ means the algorithm does not know $p$, but has sample access to independent draws of a Bernoulli random variable with mean equal to $p$. In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector $x\in P$ for a given polytope $P\subset [0,1]^n$, output a randomized vertex such that the expected value of the $i$-th coordinate is \emph{exactly} equal to $x_i$. For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix $[x_{ij}]$ and asked to sample a matching such that the probability of each edge $(i,j)$ be present in the matching is exactly equal to $x_{ij}$. We show that a polytope $P$ admits a Bernoulli factory if and and only if $P$ is the intersection of $[0,1]^n$ with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on $P$. The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings. We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the $k$-uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.
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