This article studies the problem of reconstructing the topology of a network of interacting agents via observations of the state-evolution of the agents. We focus on the large-scale network setting with the additional constraint of @math observations, where only a small fraction of the agents can be feasibly observed. The goal is to infer the underlying subnetwork of interactions and we refer to this problem as @math @math . In order to study the large-scale setting, we adopt a proper stochastic formulation where the unobserved part of the network is modeled as an Erd " o s-R 'enyi random graph, while the observable subnetwork is left arbitrary. The main result of this work is establishing that, under this setting, local tomography is actually possible with high probability, provided that certain conditions on the network model are met (such as stability and symmetry of the network combination matrix). Remarkably, such conclusion is established under the @math - @math @math , where the cardinality of the observable subnetwork is fixed, while the size of the overall network can increase without bound.
The problem of structure estimation in graphical models with latent variables is considered. We characterize conditions for tractable graph estimation and develop efficient methods with provable guarantees. We consider models where the underlying Markov graph is locally tree-like, and the model is in the regime of correlation decay. For the special case of the Ising model, the number of samples n required for structural consistency of our method scales as n=Ω(θ^(−δη(η+1)−2)_(min)log p), where p is the number of variables, θ_(min) is the minimum edge potential, δ is the depth (i.e., distance from a hidden node to the nearest observed nodes), and η is a parameter which depends on the bounds on node and edge potentials in the Ising model. Necessary conditions for structural consistency under any algorithm are derived and our method nearly matches the lower bound on sample requirements. Further, the proposed method is practical to implement and provides flexibility to control the number of latent variables and the cycle lengths in the output graph. We consider the problem of high-dimensional Gaussian graphical model selection. We identify a set of graphs for which an efficient estimation algorithm exists, and this algorithm is based on thresholding of empirical conditional covariances. Under a set of transparent conditions, we establish structural consistency (or sparsistency) for the proposed algorithm, when the number of samples n = Ω(Jmin-2 log p), where p is the number of variables and Jmin is the minimum (absolute) edge potential of the graphical model. The sufficient conditions for sparsistency are based on the notion of walk-summability of the model and the presence of sparse local vertex separators in the underlying graph. We also derive novel non-asymptotic necessary conditions on the number of samples required for sparsistency. Suppose we observe samples of a subset of a collection of random variables. No additional information is provided about the number of latent variables, nor of the relationship between the latent and observed variables. Is it possible to discover the number of latent components, and to learn a statistical model over the entire collection of variables? We address this question in the setting in which the latent and observed variables are jointly Gaussian, with the conditional statistics of the observed variables conditioned on the latent variables being specified by a graphical model. As a first step we give natural conditions under which such latent-variable Gaussian graphical models are identifiable given marginal statistics of only the observed variables. Essentially these conditions require that the conditional graphical model among the observed variables is sparse, while the effect of the latent variables is "spread out" over most of the observed variables. Next we propose a tractable convex program based on regularized maximum-likelihood for model selection in this latent-variable setting; the regularizer uses both the @math norm and the nuclear norm. Our modeling framework can be viewed as a combination of dimensionality reduction (to identify latent variables) and graphical modeling (to capture remaining statistical structure not attributable to the latent variables), and it consistently estimates both the number of latent components and the conditional graphical model structure among the observed variables. These results are applicable in the high-dimensional setting in which the number of latent observed variables grows with the number of samples of the observed variables. The geometric properties of the algebraic varieties of sparse matrices and of low-rank matrices play an important role in our analysis.