NDLI: Latent variable graphical model selection via convex optimization

Content Provider	IEEE Xplore Digital Library
Author	Chandrasekaran, V. Parrilo, P.A. Willsky, A.S.
Copyright Year	2010
Description	Author affiliation: Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, 02139, USA (Chandrasekaran, V.; Parrilo, P.A.; Willsky, A.S.)
Abstract	Suppose we have 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 hidden 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 ℓ1 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 hidden 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.
Starting Page	1610
Ending Page	1613
File Size	289571
Page Count	4
File Format	PDF
ISBN	9781424482153
e-ISBN	9781424482160
DOI	10.1109/ALLERTON.2010.5707106
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2010-09-29
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Graphical models Biological system modeling Estimation Hidden Markov models Covariance matrix Sparse matrices Matrix decomposition we refer the reader to the longer version for more details This paper is a short summary of a longer submission [8]
Content Type	Text
Resource Type	Article

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