Let @math be an undirected graph with @math edges and @math vertices. We show that @math -dimensional Ising models on @math can be learned from @math i.i.d. samples within expected total variation distance some constant factor of @math , and that this rate is optimal. We show that the same rate holds for the class of @math -dimensional multivariate normal undirected graphical models with respect to @math . We also identify the optimal rate of @math for Ising models with no external magnetic field.
Daskalakis, Dikkala, and Kamath @cite_26 were the first to study Ising models from a statistical point of view. However, their goal is to test whether an Ising model has certain properties, rather than estimating the model, which is our goal. Moreover, their focus is on designing efficient testing algorithms. They prove polynomial sample complexities and running times for testing various properties of the model.
@cite_26
2079416076 2169388387 2963904935 1499910526
The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on ( Z ^2 ) everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems. Determining the structure of dependencies among a set of variables is a common task in many signal and image processing applications, including multitarget tracking and computer vision. In this paper, we present an information-theoretic, machine learning approach to problems of this type. We cast this problem as a hypothesis test between factorizations of variables into mutually independent subsets. We show that the likelihood ratio can be written as sums of two sets of Kullback-Leibler (KL) divergence terms. The first set captures the structure of the statistical dependencies within each hypothesis, whereas the second set measures the details of model differences between hypotheses. We then consider the case when the signal prior models are unknown, so that the distributions of interest must be estimated directly from data, showing that the second set of terms is (asymptotically) negligible and quantifying the loss in hypothesis separability when the models are completely unknown. We demonstrate the utility of nonparametric estimation methods for such problems, providing a general framework for determining and distinguishing between dependency structures in highly uncertain environments. Additionally, we develop a machine learning approach for estimating lower bounds on KL divergence and mutual information from samples of high-dimensional random variables for which direct density estimation is infeasible. We present empirical results in the context of three prototypical applications: association of signals generated by sources possessing harmonic behavior, scene correspondence using video imagery, and detection of coherent behavior among sets of moving objects. Schelling’s models of segregation, first described in 1969 (Am Econ Rev 59:488–493, 1969) are among the best known models of self-organising behaviour. Their original purpose was to identify mechanisms of urban racial segregation. But his models form part of a family which arises in statistical mechanics, neural networks, social science, and beyond, where populations of agents interact on networks. Despite extensive study, unperturbed Schelling models have largely resisted rigorous analysis, prior results generally focusing on variants in which noise is introduced into the dynamics, the resulting system being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory (Young in Individual strategy and social structure: an evolutionary theory of institutions, Princeton University Press, Princeton, 1998). A series of recent papers ( in: Proceedings of the 44th annual ACM symposium on theory of computing (STOC 2012), 2012); in: 55th annual IEEE symposium on foundations of computer science, Philadelphia, 2014, J Stat Phys 158:806–852, 2015), has seen the first rigorous analyses of 1-dimensional unperturbed Schelling models, in an asymptotic framework largely unknown in statistical mechanics. Here we provide the first such analysis of 2- and 3-dimensional unperturbed models, establishing most of the phase diagram, and answering a challenge from in: Proceedings of the 44th annual ACM symposium on theory of computing (STOC 2012), 2012). We discuss a one-sample location test that can be used in the case of highdimensional data. For high-dimensional data, the power of Hotelling’s test decreases when the dimension is close to the sample size. To address this loss of power, some non-exact approaches were proposed, e.g., Dempster (1958, 1960), Bai and Saranadasa (1996) and Srivastava and Du (2006). In this paper, we focus on Hotelling’s test and Dempster’s test. The comparative merits and demerits of these two tests vary according to the local parameters. In particular, we consider the situation where it is difficult to determine which test should be used, that is, where the two tests are asymptotically equivalent in terms of local power. We propose a new statistic based on the weighted averaging of Hotelling’s T 2 statistic and Dempster’s statistic that can be applied in such a situation. Our weight is determined on the basis of the maximum local asymptotic power on a restricted parameter space that induces local asymptotic equivalence between Hotelling’s test and Dempster’s test. In addition, some good asymptotic properties with respect to the local power are shown. Numerical results show that our test is more stable than Hotelling’s T 2 statistic and Dempster’s statistic in most parameter settings.