This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Divide, Conquer and Coordinate: Globally Coordinated Switching Linear Dynamical System
April 2012 (vol. 34 no. 4)
pp. 654-669
Tai-Peng Tian, Gen. Electr. Global Res. Center, Niskayuna, NY, USA
Rui Li, Gen. Electr. Global Res. Center, Niskayuna, NY, USA
S. Sclaroff, Dept. of Comput. Sci., Boston Univ., Boston, MA, USA
The goal of this work is to learn a parsimonious and informative representation for high-dimensional time series. Conceptually, this comprises two distinct yet tightly coupled tasks: learning a low-dimensional manifold and modeling the dynamical process. These two tasks have a complementary relationship as the temporal constraints provide valuable neighborhood information for dimensionality reduction and, conversely, the low-dimensional space allows dynamics to be learned efficiently. Solving these two tasks simultaneously allows important information to be exchanged mutually. If nonlinear models are required to capture the rich complexity of time series, then the learning problem becomes harder as the nonlinearities in both tasks are coupled. A divide, conquer, and coordinate method is proposed. The solution approximates the nonlinear manifold and dynamics using simple piecewise linear models. The interactions and coordinations among the linear models are captured in a graphical model. The model structure setup and parameter learning are done using a variational Bayesian approach, which enables automatic Bayesian model structure selection, hence solving the problem of overfitting. By exploiting the model structure, efficient inference and learning algorithms are obtained without oversimplifying the model of the underlying dynamical process. Evaluation of the proposed framework with competing approaches is conducted in three sets of experiments: dimensionality reduction and reconstruction using synthetic time series, video synthesis using a dynamic texture database, and human motion synthesis, classification, and tracking on a benchmark data set. In all experiments, the proposed approach provides superior performance.

[1] D. MacKay, "Bayesian Interpolation," Neural Computation, vol. 4, no. 3, pp. 415-447, 1992.
[2] R.-S. Lin, C.-B. Liu, M.-H. Yang, N. Ahuja, and S. Levinson, "Learning Nonlinear Manifolds from Time Series," Proc. European Conf. Computer Vision, pp. 245-256, 2006.
[3] V. Pavlovic, J. Rehg, and J. MacCormick, "Learning Swtiching Linear Models of Human Motion," Proc. Advances in Neural Information Processing Systems, pp. 981-987, 2000.
[4] http://www.cwi.nl/projectsdyntex/, 2011.
[5] L. Sigal and M. Black, "HumanEva: Synchronized Video and Motion Capture Dataset for Evaluation of Articulated Human Motion," Technical Report CS-06-08, Brown Univ., 2006.
[6] S. Roweis, L. Saul, and G. Hinton, "Global Coordination of Local Linear Models," Proc. Advances in Neural Information Processing Systems, vol. 14, pp. 889-896, 2001.
[7] J. Verbeek, "Learning Non-Linear Image Manifolds by Combining Local Linear Models," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 28, no. 8, pp. 1236-1250, Aug. 2006.
[8] Z. Ghahramani and G. Hinton, "The EM Algorithm for Mixtures of Factor Analyzers," technical report, Univ. of Toronto, 1996.
[9] D. Rubin and D. Thayer, "EM Algorithm for ML Factor Analysis," Psychmetrika, vol. 47, no. 1, pp. 69-76, 1982.
[10] M. Beal, "Variational Algorithms for Approximate Bayesian Inference," PhD dissertation, Univ. College London, 2003.
[11] P. Dellaportas and J. Forster, "Markov Chain Monte Carlo Model Determination for Hierarchical and Graphical Log-Linear Models," Biometrika, vol. 86, pp. 615-633, 1996.
[12] D. MacKay, Information Theory, Inference, and Learning Algorithms. Cambridge Univ. Press, 2004.
[13] D. MacKay, "Free Energy Minimization Algorithm for Decoding and Cryptanalysis," Electronics Letters, vol. 31, no. 6, pp. 446-447, 1995.
[14] D. MacKay, "Ensemble Learning and Evidence Maximization," Proc. Neural Information Processing Systems, 1995.
[15] http://www.variational-bayes.orgvbpapers.html , 2011.
[16] D. MacKay, "Ensemble Learning for Hidden Markov Models," technical report, Cavendish Laboratory, Univ. of Cambridge, 1997.
[17] Z. Ghahramani and M. Beal, "Variational Inference for Bayesian Mixture of Factor Analysers," Proc. Advances in Neural Information Processing Systems, pp. 449-455, 1999.
[18] Z. Ghahramani and M. Beal, "Propagation Algorithms for Variational Bayesian Learning," Proc. Advances in Neural Information Processing Systems, pp. 507-513, 2000.
[19] A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian Data Analysis. Chapman & Hall/CRC Press, 1995.
[20] M. Beal, Z. Ghahramani, and C. Rasmussen, "The Infinite Hidden Markov Models," Proc. Advances in Neural Information Processing Systems, pp. 577-585, 2002.
[21] E. Fox, E. Sudderth, M. Jordan, and A. Willsky, "Nonparametric Bayesian Learning of Switching Linear Dynamical Systems," Proc. Advances in Neural Information Processing Systems, 2008.
[22] E. Fox, "Bayesian Nonparametric Learning of Complex Dynamical Phenomena," PhD dissertation, Massachusetts Inst. of Tech nology, 2009.
[23] Z. Ghahramani, "Learning Dynamic Bayesian Networks," Proc. Adaptive Processing of Sequences and Data Structures, pp. 168-197, 1998.
[24] R. Li, M.-H. Yang, S. Sclaroff, and T.-P. Tian, "Monocular Tracking of 3D Human Motion with a Coordinated Mixture of Factor Analyzers," Proc. European Conf. Computer Vision, pp. 137-150, 2006.
[25] H. Rauch, F. Tung, and C. Striebel, "Maximum Likelihood Estimates of Linear Dynamic Systems," Am. Inst. of Aeronautics and Astronautics J., vol. 3, no. 8, pp. 1445-1450, 1965.
[26] L. Rabiner and B. Juang, "An Introduction to Hidden Markov Models," IEEE ASSP Magazine, vol. 3, no. 1, pp. 4-16, Jan. 1986.
[27] R. Feyman, Statistical Mechanics. Addison-Wesley, 1972.
[28] R. Li, "Simultaneous Learning of Non-Linear Manifold and Dynamical Models for High-Dimensional Time Series," PhD dissertation, Boston Univ., 2009.
[29] S.-M. Oh, A. Ranganathan, J. Rehg, and F. Dellaert, "A Variational Inference Method for Switching Linear Dynamic System," Technical Report GIT-GVU-05-16, Georgia Inst. of Tech nology, 2005.
[30] A. Rahimi, B. Recht, and T. Darrell, "Learning to Transform Time Series with a Few Examples," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 10, pp. 1759-1775, Oct. 2007.
[31] C. Sminchisescu and A. Jepson, "Generative Modeling for Continuous Non-Linearly Embedded Visual Inference," Proc. IEEE Int'l Conf. Machine Learning, pp. 96-103, 2004.
[32] T. Jitsuhiro and S. Nakamura, "Variational Bayesian Approach for Automatic Generation of HMM Topologies," Proc. IEEE Workshop Automatic Speech Recognition and Understanding, pp. 77-82, 2003.
[33] S. Siddiqi, G. Gordon, and A. Moore, "Fast State Discovery for HMM Model Selection and Learning," Proc. 11th Int'l Conf. Artificial Intelligence and Statistics, pp. 492-499, 2007.
[34] H. Singer and M. Ostendorf, "Maximum Likelihood Successive State Splitting," Proc. Int'l Conf. Acoustics, Speech, and Signal Processing, pp. 1890-1897, 1996.
[35] J. Takami and S. Sagayama, "A Successive State Splitting Algorithm for Efficient Allophone Modeling," Proc. Int'l Conf. Acoustics, Speech, and Signal Processing, pp. 573-576, 1992.
[36] R. Poppe, "Evaluating Example-Based Pose Estimation: Experiments on the HumanEva Sets," Proc. IEEE Conf. Computer Vision and Pattern Recognition Second Workshop Evaluation of Articulated Human Motion and Pose Estimation, 2007.
[37] http:/mocap.cs.cmu.edu/, 2011.
[38] A. Elgammal and C.-S. Lee, "Inferring 3D Body Pose From Silhouettes Using Activity Manifold Learning," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 681-688, 2004.
[39] K. Grochow, S. Martin, A. Hertzman, and Z. Popovic, "Style-Based Inverse Kinematics," ACM Trans. Graphics, vol. 23, pp. 522-531, 2004.
[40] T.-P. Tian, R. Li, and S. Sclaroff, "Articulated Pose Estimation in a Learned Smooth Space of Feasible Solutions," Proc. IEEE CS Computer Vision and Pattern Recognition Learning Workshop, pp. 50-57, 2005.
[41] R. Urtasun, D. Fleet, A. Hertzman, and P. Fua, "Priors for People Tracking From Small Training Sets," Proc. IEEE Int'l Conf. Computer Vision, pp. 403-410, 2005.
[42] Z. Ghahramani and S. Roweis, "Learning Nonlinear Dynamical Systems Using an EM Algorithm," Proc. Advances in Neural Information Processing Systems, pp. 599-605, 1998.
[43] L. Ralaivola and F. d'AlchéBuc, "Dynamical Modeling with Kernels for Nonlinear Time Series Prediction," Proc. Advances in Neural Information Processing Systems, pp. 129-136, 2004.
[44] M. Varsta, J. Heikkonen, J. Lampinen, and J. Milln, "Temporal Kohonen Map and the Recurrent Self-Organizing Map: Analytical and Experimental Comparison," Neural Processing Letters, vol. 13, no. 3, pp. 237-251, 2001.
[45] O. Jenkins and M. Matarić, "A Spatio-Temporal Extension to Isomap Nonlinear Dimension Reduction," Proc. IEEE Int'l Conf. Machine Learning, pp. 56-63, 2004.
[46] J. Tenenbaum, V. Silva, and J. Langford, "A Global Geometric Framework for Nonlinear Dimensionality Reduction," Science, vol. 290, no. 5500, pp. 2319-2323, 2000.
[47] K. Moon and V. Pavlovic, "Impact of Dynamics on Subspace Embedding and Tracking of Sequences," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 198-205, 2006.
[48] J. Wang, D. Fleet, and A. Hertzman, "Gaussian Process and Dynamical Models for Human Motion," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 30, no. 2, pp. 283-298, Feb. 2008.
[49] N. Lawrence, "Gaussian Process Latent Variable Models for Visualization of High Dimensional Data," Proc. Advances in Neural Information Processing Systems, vol. 16, pp. 329-336, 2004.
[50] R. Urtasun, D. Fleet, and N. Lawrence, "Topologically-Constrained Latent Variable Models," Proc. IEEE Int'l Conf. Machine Learning, 2008.
[51] J. Wang, D. Fleet, and A. Hertzmann, "Multifactor Gaussian Process Models for Style-Content Separation," Proc. IEEE Int'l Conf. Machine Learning, pp. 975-982, 2007.
[52] N. Lawrence and A. Moore, "Hierarchical Gaussian Process Latent Variable Models," Proc. IEEE Int'l Conf. Machine Learning, vol. 227, pp. 481-488, 2007.
[53] G. Taylor, G. Hinton, and S. Roweis, "Two Distributed-State Models for Generating High-Dimensional Time Series," J. Machine Learning Research, vol. 12, pp. 1025-1068, 2011.
[54] N. Dalal and B. Triggs, "Histograms of Oriented Gradients for Human Detection," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 886-893, 2005.
[55] L. Sigal, S. Bhatia, S. Roth, M. Black, and M. Isard, "Tracking Loose-Limbed People," Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 421-428, 2004.
[56] A. Jepson, D. Fleet, and T. El-Maraghi, "Robust On-Line Appearance Models for Vision Tracking," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 25, no. 10, pp. 1296-1311, Oct. 2003.

Index Terms:
video signal processing,approximation theory,Bayes methods,image classification,image motion analysis,image reconstruction,time series,tracking,human motion tracking,globally coordinated switching linear dynamical system,informative representation,high-dimensional time series,low-dimensional manifold,dynamical process modeling,complementary relationship,dimensionality reduction,nonlinear model,time series complexity,divide method,conquer method,coordinate method,nonlinear manifold approximation,dynamics approximation,piecewise linear model,graphical model,model structure setup,parameter learning,variational Bayesian approach,automatic Bayesian model structure selection,overfitting problem,inference algorithm,dimensionality reconstruction,synthetic time series,video synthesis,dynamic texture database,human motion synthesis,human motion classification,Time series analysis,Manifolds,Computational modeling,Biological system modeling,Humans,Bayesian methods,Graphical models,human motion.,Bayesian learning,nonlinear manifold,nonlinear dynamical model,dynamic texture
Citation:
Tai-Peng Tian, Rui Li, S. Sclaroff, "Divide, Conquer and Coordinate: Globally Coordinated Switching Linear Dynamical System," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 34, no. 4, pp. 654-669, April 2012, doi:10.1109/TPAMI.2011.152
Usage of this product signifies your acceptance of the Terms of Use.