How do the structure and the parameters of Gaussian tree models affect structure learning?
The problem of learning tree-structured Gaussian graphical models from i.i.d. samples is considered. The influence of the tree structure and the parameters of the Gaussian distribution on the learning rate as the number of samples increases is discussed. Specifically, the error exponent correspondin...
| Main Authors: | , , |
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| Format: | Article |
| Language: | en_US |
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Institute of Electrical and Electronics Engineers
2011
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| Online Access: | http://hdl.handle.net/1721.1/62239 https://orcid.org/0000-0003-0149-5888 |
| _version_ | 1811070640000598016 |
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| author | Tan, Vincent Yan Fu Anandkumar, Animashree 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 Tan, Vincent Yan Fu Anandkumar, Animashree Willsky, Alan S. |
| author_sort | Tan, Vincent Yan Fu |
| collection | MIT |
| description | The problem of learning tree-structured Gaussian graphical models from i.i.d. samples is considered. The influence of the tree structure and the parameters of the Gaussian distribution on the learning rate as the number of samples increases is discussed. Specifically, the error exponent corresponding to the event that the estimated tree structure differs from the actual unknown tree structure of the distribution is analyzed. Finding the error exponent reduces to a least-squares problem in the very noisy learning regime. In this regime, it is shown that universally, the extremal tree structures which maximize and minimize the error exponent are the star and the Markov chain for any fixed set of correlation coefficients on the edges of the tree. In other words, the star and the chain graphs represent the hardest and the easiest structures to learn in the class of tree-structured Gaussian graphical models. This result can also be intuitively explained by correlation decay: pairs of nodes which are far apart, in terms of graph distance, are unlikely to be mistaken as edges by the maximum-likelihood estimator in the asymptotic regime. |
| first_indexed | 2024-09-23T08:39:15Z |
| format | Article |
| id | mit-1721.1/62239 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T08:39:15Z |
| publishDate | 2011 |
| publisher | Institute of Electrical and Electronics Engineers |
| record_format | dspace |
| spelling | mit-1721.1/622392022-09-30T10:15:17Z How do the structure and the parameters of Gaussian tree models affect structure learning? Tan, Vincent Yan Fu Anandkumar, Animashree Willsky, Alan S. Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Laboratory for Information and Decision Systems Massachusetts Institute of Technology. Stochastic Systems Group Willsky, Alan S. Willsky, Alan S. Tan, Vincent Yan Fu Anandkumar, Animashree The problem of learning tree-structured Gaussian graphical models from i.i.d. samples is considered. The influence of the tree structure and the parameters of the Gaussian distribution on the learning rate as the number of samples increases is discussed. Specifically, the error exponent corresponding to the event that the estimated tree structure differs from the actual unknown tree structure of the distribution is analyzed. Finding the error exponent reduces to a least-squares problem in the very noisy learning regime. In this regime, it is shown that universally, the extremal tree structures which maximize and minimize the error exponent are the star and the Markov chain for any fixed set of correlation coefficients on the edges of the tree. In other words, the star and the chain graphs represent the hardest and the easiest structures to learn in the class of tree-structured Gaussian graphical models. This result can also be intuitively explained by correlation decay: pairs of nodes which are far apart, in terms of graph distance, are unlikely to be mistaken as edges by the maximum-likelihood estimator in the asymptotic regime. United States. Air Force Office of Scientific Research (Grant FA9550-08-1-1080) United States. Army Research Office (MURI funded through ARO Grant W911NF-06-1-0076) United States. Air Force Office of Scientific Research (MURI Grant FA9550-06-1-0324) Singapore. Agency for Science, Technology and Research 2011-04-19T19:26:48Z 2011-04-19T19:26:48Z 2010-01 2009-09 Article http://purl.org/eprint/type/ConferencePaper 978-1-4244-5870-7 978-1-4244-5871-4 INSPEC Accession Number: 11135184 http://hdl.handle.net/1721.1/62239 Tan, Vincent Y. F., Animashree Anandkumar, and Alan S. Willsky. “How Do the Structure and the Parameters of Gaussian Tree Models Affect Structure Learning?” Communication, Control, and Computing, 2009. Allerton 2009. 47th Annual Allerton Conference On. 2009. 684-691. © 2010 IEEE. https://orcid.org/0000-0003-0149-5888 en_US http://dx.doi.org/10.1109/ALLERTON.2009.5394929 47th Annual Allerton Conference on Communication, Control, and Computing, 2009. Allerton 2009 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 Electrical and Electronics Engineers IEEE |
| spellingShingle | Tan, Vincent Yan Fu Anandkumar, Animashree Willsky, Alan S. How do the structure and the parameters of Gaussian tree models affect structure learning? |
| title | How do the structure and the parameters of Gaussian tree models affect structure learning? |
| title_full | How do the structure and the parameters of Gaussian tree models affect structure learning? |
| title_fullStr | How do the structure and the parameters of Gaussian tree models affect structure learning? |
| title_full_unstemmed | How do the structure and the parameters of Gaussian tree models affect structure learning? |
| title_short | How do the structure and the parameters of Gaussian tree models affect structure learning? |
| title_sort | how do the structure and the parameters of gaussian tree models affect structure learning |
| url | http://hdl.handle.net/1721.1/62239 https://orcid.org/0000-0003-0149-5888 |
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