A divide-and-conquer procedure for sparse inverse covariance estimation

Cho Jui Hsieh, Inderjit S. Dhillon, Pradeep Ravikumar, Arindam Banerjee

Research output: Chapter in Book/Report/Conference proceedingConference contribution

34 Scopus citations

Abstract

We consider the composite log-determinant optimization problem, arising from the ℓ1 regularized Gaussian maximum likelihood estimator of a sparse inverse covariance matrix, in a high-dimensional setting with a very large number of variables. Recent work has shown this estimator to have strong statistical guarantees in recovering the true structure of the sparse inverse covariance matrix, or alternatively the underlying graph structure of the corresponding Gaussian Markov Random Field, even in very high-dimensional regimes with a limited number of samples. In this paper, we are concerned with the computational cost in solving the above optimization problem. Our proposed algorithm partitions the problem into smaller sub-problems, and uses the solutions of the sub-problems to build a good approximation for the original problem. Our key idea for the divide step to obtain a sub-problem partition is as follows: we first derive a tractable bound on the quality of the approximate solution obtained from solving the corresponding sub-divided problems. Based on this bound, we propose a clustering algorithm that attempts to minimize this bound, in order to find effective partitions of the variables. For the conquer step, we use the approximate solution, i.e., solution resulting from solving the sub-problems, as an initial point to solve the original problem, and thereby achieve a much faster computational procedure.

Original languageEnglish (US)
Title of host publicationAdvances in Neural Information Processing Systems 25
Subtitle of host publication26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012
Pages2330-2338
Number of pages9
StatePublished - Dec 1 2012
Event26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012 - Lake Tahoe, NV, United States
Duration: Dec 3 2012Dec 6 2012

Publication series

NameAdvances in Neural Information Processing Systems
Volume3
ISSN (Print)1049-5258

Other

Other26th Annual Conference on Neural Information Processing Systems 2012, NIPS 2012
Country/TerritoryUnited States
CityLake Tahoe, NV
Period12/3/1212/6/12

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