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Published March 2010 | Published
Journal Article Open

Gaussian Multiresolution Models: Exploiting Sparse Markov and Covariance Structure


In this paper, we consider the problem of learning Gaussian multiresolution (MR) models in which data are only available at the finest scale, and the coarser, hidden variables serve to capture long-distance dependencies. Tree-structured MR models have limited modeling capabilities, as variables at one scale are forced to be uncorrelated with each other conditioned on other scales. We propose a new class of Gaussian MR models in which variables at each scale have sparse conditional covariance structure conditioned on other scales. Our goal is to learn a tree-structured graphical model connecting variables across scales (which translates into sparsity in inverse covariance), while at the same time learning sparse structure for the conditional covariance (not its inverse) within each scale conditioned on other scales. This model leads to an efficient, new inference algorithm that is similar to multipole methods in computational physics. We demonstrate the modeling and inference advantages of our approach over methods that use MR tree models and single-scale approximation methods that do not use hidden variables.

Additional Information

© 2010 IEEE. Manuscript received April 24, 2009; accepted September 22, 2009. First published November 06, 2009; current version published February 10, 2010. This work was supported in part by AFOSR under Grant FA9550-08-1-1080, in part by MURI under AFOSR Grant FA9550-06-1-0324, and in part by Shell International Exploration and Production, Inc. The work of M. J. Choi was supported in part by a Samsung Scholarship. A preliminary version of this work appeared in the Proceedings of the 26th Annual International Conference on Machine Learning (ICML 2009), Montreal, QC, Canada. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Mark J. Coates. The authors are with the Department of Electrical Engineering and Computer Science, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. The authors would like to thank Prof. H. Chen for helpful discussions about the stock returns example.

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