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Testing Ising Models

机译：测试Ising模型

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摘要

Given samples from an unknown multivariate distribution p, is it possible to distinguish whether p is the product of its marginals versus p being far from every product distribution? Similarly, is it possible to distinguish whether p equals a given distribution q versus p and q being far from each other? These problems of testing independence and goodnessof-fit have received enormous attention in statistics, information theory, and theoretical computer science, with sample-optimal algorithms known in several interesting regimes of parameters. Unfortunately, it has also been understood that these problems become intractable in large dimensions, necessitating exponential sample complexity. Motivated by the exponential lower bounds for general distributions as well as the ubiquity of Markov random fields (MRFs) in the modeling of high-dimensional distributions, we initiate the study of distribution testing on structured multivariate distributions, and in particular, the prototypical example of MRFs: the Ising Model. We demonstrate that, in this structured setting, we can avoid the curse of dimensionality, obtaining sample, and time efficient testers for independence and goodness-of-fit. One of the key technical challenges we face along the way is bounding the variance of functions of the Ising model.
机译：给定来自未知多元分布p的样本，是否可以区分p是其边际乘积与p远离每个乘积分布的乘积？同样，是否有可能区分p是否等于给定的分布q与p和q彼此相距较远？这些测试独立性和拟合优度的问题在统计学，信息论和理论计算机科学中受到了极大的关注，其中一些最优的参数体系中的样本优化算法是众所周知的。不幸的是，还已经理解，这些问题在大尺寸上变得棘手，从而需要指数级的样品复杂性。出于一般分布的指数下界以及高维分布建模中普遍存在的马尔可夫随机场（MRF）的影响，我们着手研究结构化多元分布的分布测试，尤其是关于MRF：伊辛模型。我们证明，在这种结构化的设置中，我们可以避免维数的诅咒，获取样本以及节省时间的独立性和拟合优度测试人员。我们在此过程中面临的主要技术挑战之一是限制Ising模型的功能差异。

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