Latent variable graphical model selection via convex optimization
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 latentvariable 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 maximumlikelihood for model selection in this latentvariable 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 highdimensional 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 lowrank matrices play an important role in our analysis.
 Publication:

arXiv eprints
 Pub Date:
 August 2010
 arXiv:
 arXiv:1008.1290
 Bibcode:
 2010arXiv1008.1290C
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Optimization and Control
 EPrint:
 Published in at http://dx.doi.org/10.1214/11AOS949 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)