High-dimensional structure estimation in ising models: Local separation criterion

Annals of Statistics

Volumn 40, Issue 3, 2012, Pages 1346-1375

  Anandkumar, Animashree ^a   Tan, Vincent Y F ^b   Huang, Furong ^a   Willsky, Alan S ^c  

^a Irvine   (United States)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^c Stata Center   (United States)

Author keywords

Graphical model selection; Ising models; Local separation property

Indexed keywords

EID: 84869152679     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/12-AOS1009     Document Type: Article

References (52)

1. ABBEEL, P., KOLLER, D. and NG, A. Y. (2006). Learning factor graphs in polynomial time and sample complexity. J. Mach. Learn. Res. 7 1743-1788. MR2274423
2. ALBERT, R. and BARABÁSI, A.-L. (2002). Statistical mechanics of complex networks. Rev. Modern Phys. 74 47-97. MR1895096
3. ANANDKUMAR, A., TAN, V. Y. F., HUANG, F. and WILLSKY, A. S. (2011). Highdimensional Gaussian graphical model selection: Tractable graph families. Preprint. Available at arXiv:1107.1270.
4. ANANDKUMAR, A., TAN, V. Y. F., HUANG, F. and WILLSKY, A. S. (2012). Supplement to "High-dimensional structure learning of Ising models: Local separation criterion." DOI:10.1214/12-AOS1009SUPP.
5. BAYATI, M.,MONTANARI, A. and SABERI, A. (2009). Generating random graphs with large girth. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms 566-575. SIAM, Philadelphia, PA. MR2809261
6. BENTO, J. and MONTANARI, A. (2009). Which graphical models are difficult to learn? In Proc. of Neural Information Processing Systems (NIPS).
7. BOGDANOV, A., MOSSEL, E. and VADHAN, S. (2008). The complexity of distinguishing Markov random fields. In Approximation, Randomization and Combinatorial Optimization. Lecture Notes in Comput. Sci. 5171 331-342. Springer, Berlin. MR2538798
8. BOLLOBÁS, B. (1985). Random Graphs. Academic Press, London. MR0809996
9. BRÉMAUD, P. (1999). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Texts in Applied Mathematics 31. Springer, New York. MR1689633
10. BRESLER, G.,MOSSEL, E. and SLY, A. (2008). Reconstruction ofMarkov random fields from samples: Some observations and algorithms. In Approximation, Randomization and Combinatorial Optimization. Lecture Notes in Computer Science 5171 343-356. Springer, Berlin. MR2538799
11. CHANDRASEKARAN, V., PARRILO, P. A. andWILLSKY, A. S. (2010). Latent variable graphical model selection via convex optimization. Ann. Statist. To appear. Preprint. Available on ArXiv.
12. CHECHETKA, A. and GUESTRIN, C. (2007). Efficient principled learning of thin junction trees. In Advances in Neural Information Processing Systems (NIPS).
13. CHENG, J., GREINER, R., KELLY, J., BELL, D. and LIU, W. (2002). Learning Bayesian networks from data: An information-Theory based approach. Artificial Intelligence 137 43-90. MR1906473
14. CHOI, M. J., LIM, J. J., TORRALBA, A. andWILLSKY, A. S. (2010). Exploiting hierarchical context on a large database of object categories. In IEEE Conf. on Computer Vision and Pattern Recognition (CVPR).
15. CHOI, M. J., TAN, V. Y. F., ANANDKUMAR, A. andWILLSKY, A. S. (2011). Learning latent tree graphical models. J. Mach. Learn. Res. 12 1771-1812. MR2813153
16. CHOW, C. and LIU, C. (1968). Approximating Discrete Probability Distributions with Dependence Trees. IEEE Tran. on Information Theory 14 462-467.
17. CHUNG, F. R. K. (1997). Spectral Graph Theory. CBMS Regional Conference Series in Mathematics 92. Published for the Conference Board of the Mathematical Sciences, Washington, DC. MR1421568
18. CHUNG, F. R. K. and LU, L. (2006). Complex Graphs and Network. Amer. Math. Soc., Providence, RI.
19. COVER, T. M. and THOMAS, J. A. (2006). Elements of Information Theory, 2nd ed. Wiley, Hoboken, NJ. MR2239987
20. DOMMERS, S.,GIARDINÀ, C. and VAN DER HOFSTAD, R. (2010). Ising models on power-law random graphs. J. Stat. Phys. 141 1-23.
21. DURBIN, R., EDDY, S. R., KROGH, A. and MITCHISON, G. (1999). Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge Univ. Press, Cambridge.
22. EPPSTEIN, D. (2000). Diameter and treewidth in minor-closed graph families. Algorithmica 27 275-291. MR1759751
23. GALAM, S. (1997). Rational group decision making: A random field Ising model at T = 0. Physica A: Statistical and Theoretical Physics 238 66-80.
24. GAMBURD, A., HOORY, S., SHAHSHAHANI, M., SHALEV, A. and VIRÁG, B. (2009). On the girth of random Cayley graphs. Random Structures Algorithms 35 100-117. MR2532876
25. GRABOWSKI, A. andKOSINSKI, R. (2006). Ising-based model of opinion formation in a complex network of interpersonal interactions. Physica A: Statistical Mechanics and Its Applications 361 651-664.
26. KALISCH, M. and BÜHLMANN, P. (2007). Estimating high-dimensional directed acyclic graphs with the PC-Algorithm. J. Mach. Learn. Res. 8 613-636.
27. KARGER, D. and SREBRO, N. (2001). Learning Markov networks: Maximum bounded treewidth graphs. In Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001) 392-401. SIAM, Philadelphia, PA. MR1958431
28. KEARNS, M. J. and VAZIRANI, U. V. (1994). An Introduction to Computational Learning Theory. MIT Press, Cambridge, MA. MR1292868
29. KLOKS, T. (1994). Only few graphs have bounded treewidth. Springer Lecture Notes in Computer Science 842 51-60.
30. LACIANA, C. E. and ROVERE, S. L. (2010). Ising-like agent-based technology diffusion model: Adoption patterns vs. seeding strategies. Physica A: Statistical Mechanics and Its Applications 390 1139-1149.
31. LAURITZEN, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. Oxford Univ. Press, New York. MR1419991
32. LEVIN, D. A., PERES, Y. and WILMER, E. L. (2008). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
33. LIU, H., XU, M., GU, H., GUPTA, A., LAFFERTY, J. and WASSERMAN, L. (2011). Forest density estimation. J. Mach. Learn. Res. 12 907-951. MR2786914
34. LIU, S., YING, L. and SHAKKOTTAI, S. (2010). Influence maximization in social networks: An ising-model-based approach. In Proc. 48th Annual Allerton Conference on Communication, Control, and Computing.
35. LOVÁSZ, L., NEUMANN LARA, V. and PLUMMER, M. (1978). Mengerian theorems for paths of bounded length. Period. Math. Hungar. 9 269-276. MR0509677
36. MCKAY, B. D.,WORMALD, N. C. andWYSOCKA, B. (2004). Short cycles in random regular graphs. Electron. J. Combin. 11 Research Paper 66, 12 pp. (electronic). MR2097332
37. MEINSHAUSEN, N. and BÜHLMANN, P. (2006). High-dimensional graphs and variable selection with the lasso. Ann. Statist. 34 1436-1462. MR2278363
38. MITLIAGKAS, I. and VISHWANATH, S. (2010). Strong information-Theoretic limits for source/model recovery. In Proc. 48th Annual Allerton Conference on Communication, Control and Computing.
39. NETRAPALLI, P., BANERJEE, S., SANGHAVI, S. and SHAKKOTTAI, S. (2010). Greedy learning of Markov network structure. In Proc. 48th Annual Allerton Conference on Communication, Control and Computing.
40. NEWMAN, M. E. J., WATTS, D. J. and STROGATZ, S. H. (2002). Random graph models of social networks. Proc. Natl. Acad. Sci. USA 99 2566-2572.
41. RAVIKUMAR, P., WAINWRIGHT, M. J. and LAFFERTY, J. (2010). High-dimensional Ising model selection using -1-regularized logistic regression. Ann. Statist. 38 1287-1319. MR2662343
42. RAVIKUMAR, P.,WAINWRIGHT, M. J., RASKUTTI, G. andYU, B. (2011). High-dimensional covariance estimation by minimizing -1-penalized log-determinant divergence. Electron. J. Stat. 5 935-980. MR2836766
43. SANTHANAM, N. P. and WAINWRIGHT, M. J. (2008). Information-Theoretic limits of highdimensional model selection. In International Symposium on Information Theory.
44. SPIRTES, P. and MEEK, C. (1995). Learning Bayesian networks with discrete variables from data. In Proc. of Intl. Conf. on Knowledge Discovery and Data Mining 294-299.
45. TAN, V. Y. F., ANANDKUMAR, A., TONG, L. andWILLSKY, A. S. (2011). A large-deviation analysis of the maximum-likelihood learning of Markov tree structures. IEEE Trans. Inform. Theory 57 1714-1735. MR2815845
46. TAN, V. Y. F., ANANDKUMAR, A. andWILLSKY, A. S. (2010). Learning Gaussian tree models: Analysis of error exponents and extremal structures. IEEE Trans. Signal Process. 58 2701-2714. MR2789417
47. TAN, V. Y. F., ANANDKUMAR, A. andWILLSKY, A. S. (2011). Learning high-dimensional Markov forest distributions: Analysis of error rates. J. Mach. Learn. Res. 12 1617-1653. MR2813149
48. VEGA-REDONDO, F. (2007). Complex Social Networks. Econometric Society Monographs 44. Cambridge Univ. Press, Cambridge. MR2361122
49. WAINWRIGHT, M. J. and JORDAN, M. I. (2008). Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning 1 1-305.
50. WANG, W., WAINWRIGHT, M. J. and RAMCHANDRAN, K. (2010). Information-Theoretic bounds on model selection for Gaussian Markov random fields. In IEEE International Symposium on Information Theory Proceedings (ISIT).
51. WATTS, D. J. and STROGATZ, S. H. (1998). Collective dynamics of 'small-world' networks. Nature 393 440-442.
52. GRAPHICAL MODEL OF SENATE VOTING. http://www.eecs.berkeley.edu/~elghaoui/ StatNews/ex-senate.html.