Embedded trees: Estimation of Gaussian processes on graphs with cycles

IEEE Transactions on Signal Processing

Volumn 52, Issue 11, 2004, Pages 3136-3150

  Sudderth, Erik B ^a   Wainwright, Martin J ^b   Willsky, Alan S ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; CONVERGENCE OF NUMERICAL METHODS; INFERENCE ENGINES; ITERATIVE METHODS; MARKOV PROCESSES; MATHEMATICAL MODELS; TREES (MATHEMATICS);

BELIEF PROPAGATION; ERROR VARIANCES; GAUSSIAN PROCESSES; GRAPHICAL MODELS;

SIGNAL PROCESSING;

EID: 7544249120     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2004.836539     Document Type: Article

References (39)

1. M. J. Wainwright, E. B. Sudderth, and A. S. Willsky, "Tree-based modeling and estimation of Gaussian processes on graphs with cycles," in Neural Information Processing Systems 13. Cambridge, MA: MIT Press, 2001, pp. 661-667.
2. E. B. Sudderth, "Embedded trees: Estimation of Gaussian processes on graphs with cycles," Master's thesis, Mass. Inst. Technol., Feb. 2002. [Online] Available: http://ssg.mit.edu/~esuddert/.
3. R. Szeliski, "Bayesian modeling of uncertainty in low-level vision," Int. J. Comput. Vision, vol. 5. no. 3, pp. 271-301, 1990.
4. M. R. Luettgen, W. C. Karl, and A. S. Willsky, "Efficient multiscale regularization with applications to the computation of optical flow," IEEE Trans. Image Processing, vol. 3, pp. 41-64, Jan. 1994.
5. P. W. Fieguth, W. C. Karl, A. S. Willsky, and C. Wunsch, " Multiresolution optimal interpolation and statistical analysis of TOPEX /POSEIDON satellite altimetry," IEEE Trans. Geosci. Remote Sens., vol. 33, pp. 280-292, Mar. 1995.
6. T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation. Englewood Cliffs, NJ: Prentice-Hall, 2000.
7. M. Jordan, "Graphical models," Statistical Sci., vol. 19, pp. 140-155, 2004.
8. K. C. Chou, A. S. Willsky, and A. Benveniste, "Multiscale recursive estimation, data fusion, and regularization," IEEE Trans. Automat. Contr., vol. 39, no. 3, pp. 464-478, Mar. 1994.
9. N. A. C. Cressie, Statistics for Spatial Data. New York: Wiley, 1993.
10. P. Abrahamsen, "A review of Gaussian random fields and correlation functions," Norwegian Computing Center, Oslo, Norway, Tech. Rep. 917, Apr. 1997.
11. T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991.
12. A. S. Willsky, "Multiresolution Markov models for signal and image processing," Proc. IEEE. vol. 90, pp. 1396-1458, Aug. 2002.
13. M. M. Daniel and A. S. Willsky, "A multiresolution methodology for signal-level fusion and data assimilation with applications to remote sensing," Proc. IEEE, vol. 85, pp. 164-180, Jan. 1997.
14. J. Pearl, Probabilistic Reasoning in Intelligent Systems. San Mateo, CA: Morgan Kaufman, 1988.
15. J. S. Yedidia, W. T. Freeman, and Y. Weiss, "Understanding belief propagation and its generalizations," in Exploring Artificial Intelligence in the New Millennium, G. Lakemeyer and B. Nebel, Eds. San Mateo, CA: Morgan Kaufmann, 2002.
16. S. L. Lauritzen, Graphical Models. Oxford, U.K.: Oxford Univ. Press, 1996.
17. A. P. Dempster, "Covariance selection," Biometrics, vol. 28, pp. 157-175, Mar. 1972.
18. T. P. Speed and H. T. Kiiveri, "Gaussian Markov distributions over finite graphs," Ann. Statist., vol. 14. no. 1, pp. 138-150, Mar. 1986.
19. F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, "Factor graphs and the sum-product algorithm," IEEE Trans. Inform. Theory, vol. 47, pp. 498-519, Feb. 2001.
20. S. M. Aji and R. J. McEliece, "The generalized distributive law," IEEE Trans. Inform. Theory, vol. 46, pp. 325-343, Mar. 2000.
21. Y. Weiss and W. T. Freeman, "Correctness of belief propagation in Gaussian graphical models of arbitrary topology," Neural Comput., vol. 13, pp. 2173-2200, 2001.
22. K. P. Murphy, Y. Weiss. and M. I. Jordan, "Loopy belief propagation for approximate inference: an empirical study," in Uncertainty in Artificial Intelligence 15. San Mateo, CA: Morgan Kaufmann, 1999. pp. 467-475.
23. P. Rusmevichientong and B. Van Roy, "An analysis of belief propagation on the turbo decoding graph with Gaussian densities," IEEE Trans. Inform. Theory, vol. 47, pp. 745-765, Feb. 2001.
24. J. S. Yedidia, W. T. Freeman, and Y. Weiss, "Constructing free energy approximations and generalized belief propagation algorithms," MERL, Tech. Rep. 2004-040, May 2004.
25. M. J. Wainwright, T. S. Jaakkola, and A. S. Willsky, "Tree-based reparameterization framework for analysis of sum-product and related algorithms," IEEE Trans. Inform. Theory, vol. 49, pp. 1120-1146, May 2003.
26. M. Welling and Y. W. Teh, "Linear response algorithms for approximate inference in graphical models," Neural Comput., vol. 16, pp. 197-221, 2004.
27. K. H. Plarre and P. R. Kumar, "Extended message passing algorithm for inference in loopy Gaussian graphical models," Ad Hoc Networks, vol. 2, pp. 153-169, 2004.
28. J. W. Demmel, Applied Numerical Linear Algebra. Philadelphia, PA: SIAM, 1997.
29. D. M. Young, Iterative Solution of Large Linear Systems. New York: Academic, 1971.
30. R. Barrett et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Philadelphia, PA: SIAM, 1994.
31. J. G. Berryman, "Analysis of approximate inverses in tomography II: iterative inverses," Optimiz. Eng., vol. 1, pp. 437-473, 2000.
32. D. W. Peaceman and H. H. Rachford Jr., "The numerical solution of parabolic and elliptic differential equations," J. SIAM, vol. 3, no. 1, pp. 28-41, Mar. 1955.
33. R. Nikoukhah, A. S. Willsky, and B. C. Levy, "Kalman filtering and Riccati equations for descriptor systems," IEEE Trans. Automat. Contr., vol. 37, no. 9, pp. 1325-1342, Sept. 1992.
34. L. Adams, "m-Step preconditioned conjugate gradient methods," SIAM J. Sci. Statist. Comput., vol. 6, pp. 452-463, Apr. 1985.
35. O. Axelsson, "Bounds of eigenvalues of preconditioned matrices," SIAM J. Matrix Anal. Applicat., vol. 13, no. 3, pp. 847-862, July 1992.
36. M. Bern. J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo, " Support-graph preconditioners," SIAM J. Matrix Anal. Applicat., Jan. 2001, submitted for publication.
37. D. Chen and S. Toledo, "Vaidya's preconditioners: implementation and experimental study," Electron. Trans. Numer. Anal., vol. 16, pp. 30-49, 2003.
38. E. Boman and B. Hendrickson, "Support theory for preconditioning," SIAM J. Matrix Anal. Applicat., vol. 25. no. 3. pp. 694-717, 2003.
39. E. Boman. D. Chen. B. Hendrickson, and S. Toledo, " Maximum-weight-basis preconditioners," Numerical Linear Algebra Applicat., to be published.