Open Access
June 2017 Support consistency of direct sparse-change learning in Markov networks
Song Liu, Taiji Suzuki, Raissa Relator, Jun Sese, Masashi Sugiyama, Kenji Fukumizu
Ann. Statist. 45(3): 959-990 (June 2017). DOI: 10.1214/16-AOS1470


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 $n_{p}=\Omega(d^{2}\log\frac{m^{2}+m}{2})$ and $n_{q}=\Omega({n_{p}^{2}})$, with an exponentially decaying upper-bound on learning error. Such sample complexity can be improved to $\min(n_{p},n_{q})=\Omega(d^{2}\log\frac{m^{2}+m}{2})$ 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.


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Song Liu. Taiji Suzuki. Raissa Relator. Jun Sese. Masashi Sugiyama. Kenji Fukumizu. "Support consistency of direct sparse-change learning in Markov networks." Ann. Statist. 45 (3) 959 - 990, June 2017.


Received: 1 December 2015; Revised: 1 April 2016; Published: June 2017
First available in Project Euclid: 13 June 2017

zbMATH: 1371.62022
MathSciNet: MR3662445
Digital Object Identifier: 10.1214/16-AOS1470

Primary: 62F12
Secondary: 68T99

Keywords: change detection , density ratio estimation , Markov networks

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 3 • June 2017
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