Latent variable graphical model selection via convex optimization
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 statis...
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Institute of Mathematical Statistics
2013
|
| Online Access: | http://hdl.handle.net/1721.1/77885 https://orcid.org/0000-0003-0149-5888 |
| _version_ | 1826204027301396480 |
|---|---|
| author | Willsky, Alan S. |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Willsky, Alan S. |
| author_sort | Willsky, Alan S. |
| collection | MIT |
| description | 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 ℓ[subscript 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. |
| first_indexed | 2024-09-23T12:47:41Z |
| format | Article |
| id | mit-1721.1/77885 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:47:41Z |
| publishDate | 2013 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/778852022-10-01T11:08:47Z Latent variable graphical model selection via convex optimization Willsky, Alan S. Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Willsky, Alan S. 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 ℓ[subscript 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. United States. Air Force Office of Scientific Research (AFOSR FA9550-08-1-0180) United States. Air Force Office of Scientific Research (Grant FA9550-06-1-0303) National Science Foundation (U.S.) (FRG 0757207) 2013-03-13T17:50:55Z 2013-03-13T17:50:55Z 2012 2011-11 Article http://purl.org/eprint/type/JournalArticle 0090-5364 http://hdl.handle.net/1721.1/77885 Chandrasekaran, Venkat, Pablo A. Parrilo, and Alan S. Willsky. “Latent Variable Graphical Model Selection via Convex Optimization.” The Annals of Statistics 40.4 (2012): 1935–1967. ©2012 Institute of Mathematical Statistics https://orcid.org/0000-0003-0149-5888 en_US http://dx.doi.org/10.1214/12-aos1020 Annals of Statistics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Institute of Mathematical Statistics Institute of Mathematical Statistics |
| spellingShingle | Willsky, Alan S. Latent variable graphical model selection via convex optimization |
| title | Latent variable graphical model selection via convex optimization |
| title_full | Latent variable graphical model selection via convex optimization |
| title_fullStr | Latent variable graphical model selection via convex optimization |
| title_full_unstemmed | Latent variable graphical model selection via convex optimization |
| title_short | Latent variable graphical model selection via convex optimization |
| title_sort | latent variable graphical model selection via convex optimization |
| url | http://hdl.handle.net/1721.1/77885 https://orcid.org/0000-0003-0149-5888 |
| work_keys_str_mv | AT willskyalans latentvariablegraphicalmodelselectionviaconvexoptimization |