Group Sparse Priors for Covariance Estimation

Benjamin Marlin, Mark Schmidt, Kevin Murphy
Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence, PMLR R7:391-400, 2009.

Abstract

Recently it has become popular to learn sparse Gaussian graphical models (GGMs) by imposing l1 or group l1,2 penalties on the elements of the precision matrix. Thispenalized likelihood approach results in a tractable convex optimization problem. In this paper, we reinterpret these results as performing MAP estimation under a novel prior which we call the group l1 and l1,2 positivedefinite matrix distributions. This enables us to build a hierarchical model in which the l1 regularization terms vary depending on which group the entries are assigned to, which in turn allows us to learn block structured sparse GGMs with unknown group assignments. Exact inference in this hierarchical model is intractable, due to the need to compute the normalization constant of these matrix distributions. However, we derive upper bounds on the partition functions, which lets us use fast variational inference (optimizing a lower bound on the joint posterior). We show that on two real world data sets (motion capture and financial data), our method which infers the block structure outperforms a method that uses a fixed block structure, which in turn outperforms baseline methods that ignore block structure.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR7-marlin09a, title = {Group Sparse Priors for Covariance Estimation}, author = {Marlin, Benjamin and Schmidt, Mark and Murphy, Kevin}, booktitle = {Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence}, pages = {391--400}, year = {2009}, editor = {Bilmes, Jeff and Ng, Andrew Y.}, volume = {R7}, series = {Proceedings of Machine Learning Research}, month = {18--21 Jun}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r7/main/assets/marlin09a/marlin09a.pdf}, url = {https://proceedings.mlr.press/r7/marlin09a.html}, abstract = {Recently it has become popular to learn sparse Gaussian graphical models (GGMs) by imposing l1 or group l1,2 penalties on the elements of the precision matrix. Thispenalized likelihood approach results in a tractable convex optimization problem. In this paper, we reinterpret these results as performing MAP estimation under a novel prior which we call the group l1 and l1,2 positivedefinite matrix distributions. This enables us to build a hierarchical model in which the l1 regularization terms vary depending on which group the entries are assigned to, which in turn allows us to learn block structured sparse GGMs with unknown group assignments. Exact inference in this hierarchical model is intractable, due to the need to compute the normalization constant of these matrix distributions. However, we derive upper bounds on the partition functions, which lets us use fast variational inference (optimizing a lower bound on the joint posterior). We show that on two real world data sets (motion capture and financial data), our method which infers the block structure outperforms a method that uses a fixed block structure, which in turn outperforms baseline methods that ignore block structure.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Group Sparse Priors for Covariance Estimation %A Benjamin Marlin %A Mark Schmidt %A Kevin Murphy %B Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2009 %E Jeff Bilmes %E Andrew Y. Ng %F pmlr-vR7-marlin09a %I PMLR %P 391--400 %U https://proceedings.mlr.press/r7/marlin09a.html %V R7 %X Recently it has become popular to learn sparse Gaussian graphical models (GGMs) by imposing l1 or group l1,2 penalties on the elements of the precision matrix. Thispenalized likelihood approach results in a tractable convex optimization problem. In this paper, we reinterpret these results as performing MAP estimation under a novel prior which we call the group l1 and l1,2 positivedefinite matrix distributions. This enables us to build a hierarchical model in which the l1 regularization terms vary depending on which group the entries are assigned to, which in turn allows us to learn block structured sparse GGMs with unknown group assignments. Exact inference in this hierarchical model is intractable, due to the need to compute the normalization constant of these matrix distributions. However, we derive upper bounds on the partition functions, which lets us use fast variational inference (optimizing a lower bound on the joint posterior). We show that on two real world data sets (motion capture and financial data), our method which infers the block structure outperforms a method that uses a fixed block structure, which in turn outperforms baseline methods that ignore block structure. %Z Reissued by PMLR on 04 October 2026.
APA
Marlin, B., Schmidt, M. & Murphy, K.. (2009). Group Sparse Priors for Covariance Estimation. Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R7:391-400 Available from https://proceedings.mlr.press/r7/marlin09a.html. Reissued by PMLR on 04 October 2026.

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