SUPPORT CONSISTENCY OF DIRECT SPARSE-CHANGE LEARNING IN MARKOV NETWORKS

• 成果类型: Article
• 署名作者: Liu, Song; Suzuki, Taiji; Relator, Raissa; Sese, Jun; Sugiyama, Masashi; Fukumizu, Kenji
• 署名单位: Research Organization of Information & Systems (ROIS); Institute of Statistical Mathematics (ISM) - Japan; Institute of Science Tokyo; Tokyo Institute of Technology; University of Tokyo; National Institute of Advanced Industrial Science & Technology (AIST)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1470
• 发表日期: 2017
• 页码: 959-990
• 关键词: inverse covariance estimation; model selection

摘要

We study the problem of learning sparse structure changes between two Markov networks P and Q. Rather than fitting two Markov networks separately to two sets of data and figuring out their differences, a recent work proposed to learn changes directly via estimating the ratio between two Markov network models. In this paper, we give sufficient conditions for successful change detection with respect to the sample size n(p), n(q), the dimension of data m and the number of changed edges d. When using an unbounded density ratio model, we prove that the true sparse changes can be consistently identified for n(p) = Omega(d(2) log m(2)+m/2) and n(q) = Omega ( n(p)(2)), with an exponentially decaying upper- bound on learning error. Such sample complexity can be improved to min( n(p), n(q)) = Omega (d(2) log m(2) +m/2) when the boundedness of the density ratio model is assumed. Our theoretical guarantee can be applied to a wide range of discrete/continuous Markov networks.