LEARNING LOOPY GRAPHICAL MODELS WITH LATENT VARIABLES: EFFICIENT METHODS AND GUARANTEES

• 成果类型: Article
• 署名作者: Anandkumar, Animashree; Valluvan, Ragupathyraj
• 署名单位: University of California System; University of California Irvine
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS1070
• 发表日期: 2013
• 页码: 401-435
• 关键词: ising-models; selection; TREE; networks; girth

摘要

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 = Omega(theta(-delta eta(eta+1)-2)(min)log p), where p is the number of variables, theta(min) is the minimum edge potential, delta is the depth (i.e., distance from a hidden node to the nearest observed nodes), and eta 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.