The Annals of Statistics
- Ann. Statist.
- Volume 40, Number 4 (2012), 1935-1967.
Latent variable graphical model selection via convex optimization
Venkat Chandrasekaran, Pablo A. Parrilo, and Alan S. Willsky
Abstract
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 $\ell_{1}$ 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.
Article information
Source
Ann. Statist., Volume 40, Number 4 (2012), 1935-1967.
Dates
First available in Project Euclid: 30 October 2012
Permanent link to this document
https://projecteuclid.org/euclid.aos/1351602527
Digital Object Identifier
doi:10.1214/11-AOS949
Mathematical Reviews number (MathSciNet)
MR3059067
Zentralblatt MATH identifier
1288.62085
Subjects
Primary: 62F12: Asymptotic properties of estimators 62H12: Estimation
Keywords
Gaussian graphical models covariance selection latent variables regularization sparsity low-rank algebraic statistics high-dimensional asymptotics
Citation
Chandrasekaran, Venkat; Parrilo, Pablo A.; Willsky, Alan S. Latent variable graphical model selection via convex optimization. Ann. Statist. 40 (2012), no. 4, 1935--1967. doi:10.1214/11-AOS949. https://projecteuclid.org/euclid.aos/1351602527
See also
- Discussion: Latent variable graphical model selection via convex optimization. Digital Object Identifier: doi:10.1214/12-AOS979
- Discussion: Latent variable graphical model selection via convex optimization. Digital Object Identifier: doi:10.1214/12-AOS980
- Discussion: Latent variable graphical model selection via convex optimization. Digital Object Identifier: doi:10.1214/12-AOS981
- Discussion: Latent variable graphical model selection via convex optimization. Digital Object Identifier: doi:10.1214/12-AOS984
- Discussion: Latent variable graphical model selection via convex optimization. Digital Object Identifier: doi:10.1214/12-AOS985
- Discussion: Latent variable graphical model selection via convex optimization. Digital Object Identifier: doi:10.1214/12-AOS1001
- Rejoinder: Latent variable graphical model selection via convex optimization. Digital Object Identifier: doi:10.1214/12-AOS1020
Supplemental materials
- Supplementary material: Supplement to “Latent variable graphical model selection via convex optimization”. Due to space constraints, we have moved some technical proofs to a supplementary document [6].Digital Object Identifier: doi:10.1214/11-AOS949SUPP
