Faculty Publications - Martin Wainwright

Book chapters or sections

• M. Wainwright, P. Ravikumar, and J. D. Lafferty, "High-dimensional graphical model selection using l_1-regularized logistic regression," in Advances in Neural Information Processing Systems 19: Proc. 20th Annual Conf. (NIPS 2006), B. Scholkopf, J. Platt, and T. Hofmann, Eds., Neural Information Processing Systems, Vol. 19, Cambridge, MA: MIT Press, 2007, pp. 1465-1472.
• M. Wainwright and M. Jordan, "A variational principle for graphical models," in New Directions in Statistical Signal Processing: From Systems to Brain, S. Haykin, J. C. Principe, T. J. Sejnowski, and J. McWhirter, Eds., Neural Information Processing, Cambridge, MA: MIT Press, 2006, pp. 155-202.
• M. Wainwright, "Estimating the "wrong" Markov random field: Benefits in the computation-limited setting," in Advances in Neural Information Processing Systems 18: Proc. 19th Annual Conf. (NIPS 2005), Y. Weiss, B. Scholkopf, and J. Platt, Eds., Neural Information Processing Systems, Vol. 18, Cambridge, MA: MIT Press, 2006, pp. 1425-1432.
• X. Nguyen, M. Wainwright, and M. Jordan, "Divergences, surrogate loss functions and experimental design," in Advances in Neural Information Processing Systems 18: Proc. 19th Annual Conf. (NIPS 2005), Y. Weiss, B. Scholkopf, and J. Platt, Eds., Neural Information Processing Systems, Vol. 18, Cambridge, MA: MIT Press, 2006, pp. 1011-1018.
• M. Wainwright and M. Jordan, "Semidefinite relaxations for approximate inference on graphs with cycles," in Advances in Neural Information Processing Systems 16: Proc. 17th Annual Conf. (NIPS 2003), S. Thrun, L. K. Saul, and B. Schoelkopf, Eds., Neural Information Processing Systems, Vol. 16, Cambridge, MA: MIT Press, 2004, pp. 369-376.
• M. Wainwright, T. S. Jaakkola, and A. S. Willsky, "Exact MAP estimates by (hyper)tree agreement," in Advances in Neural Information Processing Systems 15: Proc. 16th Annual Conf. (NIPS 2002), S. Becker, S. Thrun, and K. Obermayer, Eds., Neural Information Processing Systems, Vol. 15, Cambridge, MA: MIT Press, 2003, pp. 833-840.
• L. Chen, M. Wainwright, M. Cetin, and A. S. Willsky, "Multitarget-multisensor data association using the tree-reweighted max-product algorithm," in Signal Processing, Sensor Fusion, and Target Recognition XII, I. Kadar, Ed., Proceedings of SPIE, Vol. 5096, Bellingham, WA: SPIE -- Society of Photo-Optical Instrumentation Engineers, 2003, pp. 127-138.
• M. Wainwright, E. B. Sudderth, and A. S. Willsky, "Tree-based modeling and estimation of Gaussian processes on graphs with cycles," in Advances in Neural Information Processing Systems 13: Proc. 14th Annual Conf. (NIPS 2000), T. K. Leen, T. G. Dietterich, and V. Tresp, Eds., Neural Information Processing Systems, Vol. 13, Cambridge, MA: MIT Press, 2001, pp. 661-667.
• M. Wainwright and E. P. Simoncelli, "Scale mixtures of Gaussians and the statistics of natural images," in Advances in Neural Information Processing Systems 12: Proc. 13th Annual Conf. (NIPS 1999), S. A. Solla, T. K. Leen, and K. R. Muller, Eds., Neural Information Processing Systems, Vol. 12, Cambridge, MA: MIT Press, 2000, pp. 855-861.

Articles in journals or magazines

• Z. Zhang, V. Anantharam, M. Wainwright, and B. Nikolic, "An Efficient 10GBASE-T Ethernet LDPC Decoder Design With Low Error Floors," Solid-State Circuits, IEEE Journal of, vol. 45, no. 4, pp. 843-855, April 2010.
• T. G. Roosta, M. Wainwright, and S. S. Sastry, "Convergence analysis of reweighted sum-product algorithms," IEEE Trans. Signal Processing, vol. 56, no. 9, pp. 4293-4305, Sep. 2008.
• C. Daskalakis, A. G. Dimakis, R. M. Karp, and M. Wainwright, "Probabilistic analysis of linear programming decoding," IEEE Trans. Information Theory, vol. 54, no. 8, pp. 3565-3578, Aug. 2008.
• X. Nguyen, M. Wainwright, and M. Jordan, "On optimal quantization rules for some problems in sequential decentralized detection," IEEE Transactions on Information Theory, vol. 54, no. 7, pp. 3285-3295, July 2008.
• A. D. G. Dimakis, A. D. Sarwate, and M. Wainwright, "Geographic gossip: Efficient averaging for sensor networks," IEEE Transactions on Signal Processing, vol. 56, no. 3, pp. 1205-1216, March 2008.
• M. Wainwright, "Sparse graph codes for side information and binning," IEEE Signal Processing Magazine, vol. 24, no. 5, pp. 47-57, Sep. 2007.
• E. Maneva, E. Mossel, and M. Wainwright, "A new look at survey propagation and its generalizations," J. ACM, vol. 54, no. 4, pp. 17, July 2007.
• J. Feldman, T. Malkin, R. A. Servedio, C. Stein, and M. Wainwright, "LP decoding corrects a constant fraction of errors," IEEE Transactions on Information Theory, vol. 53, no. 1, pp. 82-89, Jan. 2007.
• M. Wainwright, "Estimating the "wrong" graphical model: Benefits in the computation-limited setting," Journal of Machine Learning Research, vol. 7, pp. 1829--1859, Dec. 2006.
• M. Cetin, L. Chen, J. W. Fisher III, A. T. Ihler, R. L. Moses, M. Wainwright, and A. S. Willsky, "Distributed fusion in sensor networks: A graphical models perspective," IEEE Signal Processing Magazine, vol. 23, no. 4, pp. 42-55, July 2006.
• M. Wainwright and M. Jordan, "Log-determinant relaxation for approximate inference in discrete Markov random fields," IEEE Transactions on Signal Processing, vol. 54, no. 6, pt. 1, pp. 2099-2109, June 2006.
• L. Chen, M. Wainwright, M. Cetin, and A. S. Willsky, "Data association based on optimization in graphical models with application to sensor networks," Mathematical and Computer Modelling: Optimization and Control for Military Applications, vol. 43, no. 9-10, pp. 1114-1135, May 2006.
• M. Wainwright, T. S. Jaakkola, and A. S. Willsky, "MAP estimation via agreement on trees: Message-passing and linear programming," IEEE Transactions on Information Theory, vol. 51, no. 11, pp. 3697-3717, Nov. 2005.
• X. Nguyen, M. Wainwright, and M. Jordan, "Nonparametric decentralized detection using kernel methods," IEEE Transactions on Signal Processing, vol. 53, no. 11, pp. 4053-4066, Nov. 2005.
• M. Wainwright, T. S. Jaakkola, and A. S. Willsky, "A new class of upper bounds on the log partition function," IEEE Transactions on Information Theory, vol. 51, no. 7, pp. 2313-2335, July 2005.
• J. Feldman, M. Wainwright, and D. R. Karger, "Using linear programming to decode binary linear codes," IEEE Transactions on Information Theory, vol. 51, no. 3, pp. 954-972, March 2005.
• E. B. Sudderth, M. Wainwright, and A. S. Willsky, "Embedded trees: Estimation of Gaussian processes on graphs with cycles," IEEE Transactions on Signal Processing, vol. 52, no. 11, pp. 3136-3150, Nov. 2004.
• M. Wainwright, T. Jaakkola, and A. Willsky, "Tree consistency and bounds on the performance of the max-product algorithm and its generalizations," Statistics and Computing, vol. 14, no. 2, pp. 143-166, April 2004.
• J. Portilla, V. Strela, M. Wainwright, and E. P. Simoncelli, "Image denoising using scale mixtures of Gaussians in the wavelet domain," IEEE Transactions on Image Processing, vol. 12, no. 11, pp. 1338-1351, Nov. 2003.
• M. Wainwright, T. S. Jaakkola, and A. S. Willsky, "Tree-based reparameterization framework for analysis of sum-product and related algorithms," IEEE Transactions on Information Theory, vol. 49, no. 5, pp. 1120-1146, May 2003.
• M. Wainwright, E. P. Simoncelli, and A. S. Willsky, "Random cascades on wavelet trees and their use in analyzing and modeling natural images," Applied and Computational Harmonic Analysis, vol. 11, pp. 89--123, 2001.
• M. Wainwright, "Visual adaptation as optimal information transmission," Vision Research, vol. 39, pp. 3960-3974, Nov. 1999.

Articles in conference proceedings

• N. Santhanam and M. Wainwright, "Information-theoretic limits of graphical model selection in high dimensions," in Proc. 2008 IEEE Intl. Symp. on Information Theory (ISIT 2008), Piscataway, NJ: IEEE Press, 2008, pp. 2136-2140.
• D. Omidiran and M. Wainwright, "High-dimensional subset recovery in noise: Sparse measurements and statistical efficiency," in Proc. 2008 IEEE Intl. Symp. on Information Theory (ISIT 2008), Piscataway, NJ: IEEE Press, 2008, pp. 2192-2196.
• W. Wang, M. Wainwright, and K. Ramchandran, "Information-theoretic limits on sparse support recovery: Dense versus sparse measurements," in Proc. 2008 IEEE Intl. Symp. on Information Theory (ISIT 2008), Piscataway, NJ: IEEE Press, 2008, pp. 2197-2201.
• A. A. Amini and M. Wainwright, "High-dimensional analysis of semidefinite relaxations for sparse principal components," in Proc. 2008 IEEE Intl. Symp. on Information Theory (ISIT 2008), Piscataway, NJ: IEEE Press, 2008, pp. 2454-2458.
• P. Lee, L. Dolecek, Z. Zhang, V. Anantharam, B. Nikolic, and M. Wainwright, "Error floors in LDPC codes: Fast simulation, bounds and hardware emulation," in Proc. 2008 IEEE Intl. Symp. on Information Theory (ISIT 2008), Piscataway, NJ: IEEE Press, 2008, pp. 444-448.
• P. Ravikumar, A. Agarwal, and M. Wainwright, "Message-passing for graph-structured linear programs: Proximal projections, convergence and rounding schemes," in Proc. 25th Intl. Conf. on Machine Learning (ICML 2008), A. McCallum and S. Roweis, Eds., ACM International Conference Proceeding Series, Vol. 307, New York, NY: The Association for Computing Machinery, Inc., 2008, pp. 800-807.
• R. Rajagopal and M. Wainwright, "Stochastic approximation analysis of distributed estimation algorithms," in Proc. 45th Annual Allerton Conf. on Communication, Control, and Computing, C. Hadjicostis and P. Moulin, Eds., Piscataway, NJ: IEEE Press, 2007.
• E. Sudderth, M. Wainwright, and A. Willsky, "Loop series and Bethe variational bounds in attractive graphical models (Invited Paper)," in Proc. 45th Annual Allerton Conf. on Communication, Control, and Computing, C. Hadjicostis and P. Moulin, Eds., Piscataway, NJ: IEEE Press, 2007, pp. 534-534.
• L. Dolecek, Z. Zhang, M. Wainwright, V. Anantharam, and B. Nikolic, "Evaluation of the low frame error rate performance of LDPC codes using importance sampling," in Proc. 2007 IEEE Information Theory Workshop (ITW '07), Piscataway, NJ: IEEE Press, 2007, pp. 202-207.
• A. G. Dimakis, M. Wainwright, and K. Ramchandran, "Lower bounds on the rate-distortion function of LDGM codes," in Proc. 2007 IEEE Information Theory Workshop (ITW '07), Piscataway, NJ: IEEE Press, 2007, pp. 650-655.
• Z. Zhang, L. Dolecek, M. Wainwright, V. Anantharam, and B. Nikolic, "Quantization effects in low-density parity-check decoders," in Proc. 2007 IEEE Intl. Conf. on Communications (ICC '07), Piscataway, NJ: IEEE Press, 2007, pp. 6231-6237.
• L. Dolecek, Z. Zhang, V. Anantharam, M. Wainwright, and B. Nikolic, "Analysis of absorbing sets for array-based LDPC codes," in Proc. IEEE International Conference on Communications (ICC '07), Piscataway, NJ: IEEE Press, 2007, pp. 6261-6268.
• M. Wainwright, "Information-theoretic bounds on sparsity recovery in the high-dimensional and noisy setting," in Proc. 2007 IEEE International Symposium on Information Theory (ISIT 2007), Piscataway, NJ: IEEE Press, 2007, pp. 961-965.
• X. Nguyen, M. Wainwright, and M. Jordan, "Nonparametric estimation of the likelihood ratio and divergence functionals," in Proc. 2007 IEEE International Symposium on Information Theory (ISIT 2007), Piscataway, NJ: IEEE Press, 2007, pp. 2016-2020.
• A. G. Dimakis, P. B. Godfrey, M. Wainwright, and K. Ramchandran, "Network coding for distributed storage systems," in Proc. 26th IEEE International Conference on Computer Communications (INFOCOM 2007), Piscataway, NJ: IEEE Press, 2007, pp. 2000-2008.
• T. Roosta, M. Wainwright, and S. S. Sastry, "Convergence analysis of reweighted sum-product algorithms," in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2007), Vol. 2, Piscataway, NJ: IEEE Press, 2007, pp. 541-544.
• J. Schiff, D. Antonelli, A. G. Dimakis, D. Chu, and M. Wainwright, "Robust message-passing for statistical inference in sensor networks," in Proc. 6th International Symposium on Information Processing in Sensor Networks (IPSN 2007), New York, NY: ACM Press, 2007, pp. 109-118.
• K. Daskalakis, G. A. Dimakis, R. M. Karp, and M. Wainwright, "Probabilistic analysis of linear programming decoding," in Proc. 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), New York, NY/Philadelphia, PA: The Association for Computing Machinery, Inc./Society for Industrial and Applied Mathematics, 2007, pp. 385-394.
• Z. Zhang, L. Dolecek, B. Nikolic, V. Anantharam, and M. Wainwright, "Investigation of error floors of structured low-density parity-check codes by hardware emulation," in Proc. IEEE Global Telecommunications Conference (GLOBECOM '06), Piscataway, NJ: IEEE Press, 2006, pp. 1-6.
• E. Martinian and M. Wainwright, "Low-density constructions can achieve the Wyner-Ziv and Gelfand-Pinsker bounds," in Proc. 2006 IEEE International Symposium on Information Theory (ISIT 2006), Piscataway, NJ: IEEE Press, 2006, pp. 484-488.
• A. G. Dimakis and M. Wainwright, "Guessing facets: Polytope structure and improved LP decoder," in Proc. IEEE International Symposium on Information Theory (ISIT 2006), Piscataway, NJ: IEEE Press, 2006, pp. 1369-1373.
• X. Nguyen, M. Wainwright, and M. Jordan, "On optimal quantization rules for sequential decision problems," in Proc. IEEE International Symposium on Information Theory (ISIT 2006), Piscataway, NJ: IEEE Press, 2006, pp. 2652-2656.
• R. Rajagopal, M. Wainwright, and P. Varaiya, "Universal quantile estimation with feedback in the communication-constrained setting," in Proc. 2006 IEEE International Symposium on Information Theory (ISIT 2006), Piscataway, NJ: IEEE Press, 2006, pp. 836-840.
• A. G. Dimakis, A. D. Sarwate, and M. Wainwright, "Geographic gossip: Efficient aggregation for sensor networks," in Proc. IEEE 5th International Conference on Information Processing in Sensor Networks (IPSN 2006), Piscataway, NJ: IEEE Press, 2006, pp. 69-76.
• E. Martinian and M. Wainwright, "Low density codes achieve the rate-distortion bound," in Proc. Data Compression Conference (DCC 2006), J. A. Storer and M. Cohn, Eds., Los Alamitos, CA: IEEE Computer Society, 2006, pp. 153-162.
• E. Martinian and M. Wainwright, "Low-density constructions for lossy compression, binning, and coding with side information," in Proc. IEEE Information Theory Workshop 2006 (ITW '06), Piscataway, NJ: IEEE Press, 2006, pp. 263-264.
• V. Kolmogorov and M. Wainwright, "On the Optimality of Tree-reweighted Max-product Message-passing," in Proc. of the 21st Conference in Uncertainty in Artificial Intelligence (UAI '05), Arlington, Virginia: AUAI Press, 2005, pp. 316-323.
• M. Wainwright and E. Maneva, "Lossy source encoding via message-passing and decimation over generalized codewords of LDGM codes," in Proc. IEEE International Symposium on Information Theory (ISIT 2005), Piscataway, NJ: IEEE Press, 2005, pp. 1493-1497.
• X. Nguyen, M. Wainwright, and M. Jordan, "On information divergence measures, surrogate loss functions and decentralized hypothesis testing," in Proc. 43rd Annual Allerton Conference on Communication, Control, and Computing, G. Dullerud and A. Singer, Eds., Urbana-Champaign, IL: University of Illinois, 2005.
• M. Wainwright, "Convex surrogates and stable message-passing: Joint parameter estimation and prediction in coupled Gaussian mixture models," in 2005 IEEE/SP 13th Workshop on Statistical Signal Processing Book of Abstracts, Piscataway, NJ: IEEE Press, 2005, pp. 411-416.
• E. Maneva, E. Mossel, and M. Wainwright, "A new look at survey propagation and its generalizations (Extended Abstract)," in Proc. 16th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA 2005), Philadelphia, PA: Society for Industrial and Applied Mathematics, 2005, pp. 1089-1098.
• E. N. Maneva, E. Mossel, and M. Wainwright, "A new look at survey propagation and its generalizations," in Proc. 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), SIAM, 2005, pp. 1089-1098.
• M. Wainwright and M. Jordan, "Variational inference in graphical models: The view from the marginal polytope," in Proc. 41st Allerton Conference on Communication, Control, and Computing, Urbana-Champaign, IL: University of Illinois, 2004.
• J. Feldman, T. Malkin, R. A. Servedio, C. Stein, and M. Wainwright, "LP decoding corrects a constant fraction of errors," in Proc. IEEE International Symposium on Informtion Theory (ISIT 2004), Piscataway, NJ: IEEE Press, 2004, pp. 69-69.
• X. Nguyen, M. Wainwright, and M. Jordan, "Decentralized detection and classification using kernel methods," in Proc. 21st International Conference on Machine Learning (ICML '04), ACM International Conference Proceeding Series, Vol. 69, New York, NY: The Association for Computing Machinery, Inc., 2004, pp. Art. 80.
• J. Portilla, V. Strela, M. Wainwright, and E. Simoncelli, "Adaptive Wiener denoising using a Gaussian scale mixture model in the wavelet domain," in Proc. 2001 International Conference on Image Processing, Vol. 2, Piscataway, NJ: IEEE Press, 2001, pp. 37-40.
• M. Wainwright, E. P. Simoncelli, and A. S. Willsky, "Random cascades of Gaussian scale mixtures and their use in modeling natural images with application to denoising," in Proc. 2000 International Conference on Image Processing, Vol. 1, Piscataway, NJ: IEEE Press, 2000, pp. 260 - 263.

Technical Reports

• P. Ravikumar, A. Agarwal, and M. Wainwright, "Message-Passing for Graph-Structured Linear Programs: Proximal Methods and Rounding Schemes," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-08-765, Oct. 2008.
• X. L. Nguyen, M. Wainwright, and M. Jordan, "Estimating Divergence Functionals and the Likelihood Ratio by Convex Risk Minimization," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-08-764, Sep. 2008.
• G. Obozinski, M. Wainwright, and M. Jordan, "Union Support Recovery in High-Dimensional Multivariate Regression," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-08-761, Aug. 2008.
• W. Wang, M. Wainwright, and K. Ramchandran, "Information-Theoretic Limits on Sparse Signal Recovery: Dense versus Sparse Measurement Matrices," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-08-754, May 2008.
• D. Omidiran and M. Wainwright, "High-Dimensional Subset Recovery in Noise: Sparsified Measurements Without Loss of Statistical Efficiency," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-08-753, May 2008.
• R. Rajagopal, M. Wainwright, R. Rajagopal, and M. Wainwright, "Network-Based Consensus Averaging with General Noisy Channels," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-08-751, May 2008.
• P. Vavikumar, M. Wainwright, and J. D. Lafferty, "High-Dimensional Graphical Model Selection Using l_1-Regularized Logistic Regression," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-08-750, April 2008.
• A. A. Amini and M. Wainwright, "High-Dimensional Analysis of Semidefinite Relaxations for Sparse Principal Components," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-08-747, March 2008.
• T. Roosta, M. Wainwright, and S. S. Sastry, "Convergence Analysis of Reweighted Sum-Product Algorithms," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-07-737, Aug. 2007.
• M. Wainwright and E. Martinian, "Low-Density Graph Codes That Are Optimal for Source/Channel Coding and Binning," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-07-730, April 2007.
• M. Wainwright, "Information-Theoretic Limits on Sparsity Recovery in the High-Dimensional and Noisy Setting," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-07-725, Jan. 2007.
• C. Daskalakis, G. A. Dimakis, R. M. Karp, and M. Wainwright, "Probabilistic Analysis of Linear Programming Decoding," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-06-718, Oct. 2006.
• M. Wainwright, "Sharp Thresholds for High-Dimensional and Noisy Recovery of Sparsity," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-06-709, June 2006.
• X. Nguyen, M. Wainwright, and M. Jordan, "On Optimal Quantization Rules for Some Sequential Decision Problems," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-06-708, June 2006.
• X. Nguyen, M. Wainwright, and M. Jordan, "On Divergences, Surrogate Loss Functions, and Decentralized Detection," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-05-695, Oct. 2005.
• E. Maneva, E. Mossel, and M. Wainwright, "A New Look at Survey Propagation and Its Generalizations," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-04-669, Sep. 2004.
• M. Wainwright and M. Jordan, "Treewidth-Based Conditions for Exactness of the Sherali-Adams & Lasserre Relaxations," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-04-671, Sep. 2004.
• X. Nguyen, M. Wainwright, and M. Jordan, "Decentralized Detection and Classification Using Kernel Methods," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-04-658, April 2004.
• M. Wainwright and M. Jordan, "Graphical Models, Exponential Families, and Variational Inference," UC Berkeley, Department of Statistics, Tech. Rep. UCB/STAT-03-649, Sep. 2003.
• M. J. Wainwright, T. S. Jaakkola, and A. S. Willsky, "MAP Estimation via Agreement on (hyper)trees: Message-passing and Linear Programming Approaches," EECS Department, University of California, Berkeley, Tech. Rep. UCB/CSD-03-1269, Aug. 2003. [abstract]
• M. J. Wainwright and M. I. Jordan, "Semidefinite Relaxations for Approximate Inference on Graphs with Cycles," EECS Department, University of California, Berkeley, Tech. Rep. UCB/CSD-03-1226, Jan. 2003. [abstract]

Unpublished articles

• M. Wainwright, "Inconsistent parameter estimation in Markov random fields: Benefits in the computation-limited setting," Feb. 2006.

Ph.D. Theses

• M. Wainwright, "Stochastic processes on graphs with cycles: Geometric and variational approaches," Massachusetts Institute of Technology, 2002.