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Learning Dynamic Models from Unsequenced Data Jeff Schneider School of Computer Science

Learning Dynamic Models from Unsequenced Data Jeff Schneider School of Computer Science Carnegie Mellon University. joint work with Tzu-Kuo Huang, Le Song. Learning Dynamic Models. Hidden Markov Models e.g. for speech recognition Dynamic Bayesian Networks e.g. for protein/gene interaction

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Learning Dynamic Models from Unsequenced Data Jeff Schneider School of Computer Science

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  1. Learning Dynamic Models from Unsequenced Data Jeff Schneider School of Computer Science Carnegie Mellon University joint work with Tzu-Kuo Huang, Le Song

  2. Learning Dynamic Models Hidden Markov Models e.g. for speech recognition Dynamic Bayesian Networks e.g. for protein/gene interaction System Identification e.g. for control [source: Wikimedia Commons] • Key Assumption: SEQUENCED observations • What if observations are NOT SEQUENCED? [source: SISL ARLUT] Hubble Ultra Deep Field [source: UAV ETHZ] [Bagnell & Schneider, 2001]

  3. When are Observations not Sequenced? • Galaxy evolution • dynamics are too slow to watch • Slow developing diseases • Alzheimers • Parkinsons • Biological processes • measurements are often destructive [source: STAGES] [source: Getty Images] How can we learn dynamic models for these? [source: Bryan Neff Lab, UWO]

  4. Outline • Linear Models • [Huang and Schneider, ICML, 2009] • Nonlinear Models • [Huang, Song, Schneider, AISTATS, 2010] • Combining Sequence and Unsequenced Data • [Huang and Schneider, NIPS, 2011]

  5. Problem Description Estimate A from the sample of xi’s

  6. Doesn't seem impossible …

  7. Identifiability Issues

  8. Identifiability Issues

  9. A Maximum Likelihood Approach suppose we knew the dynamic model and the predecessor of each point …

  10. Likelihood continued

  11. Likelihood (continued) • we don’t know the time either so also integrate out over time • then use the empirical density as an estimate for the resulting marginal distribution

  12. Unordered Method (UM): Estimation

  13. Expectation Maximization

  14. Sample Synthetic Result input output

  15. Partial-order Method (PM)

  16. Partial Order Approximation (PM) • Perform estimation by alternating maximization • Replace UM's E-step with a maximum spanning tree on the complete graph over data points • weight on each edge is probability of one point being generated from the other given A and s • enforces a global consistency on the solution • M-step is unchanged: weighted regression

  17. Learning Nonlinear Dynamic Models [Huang, Song, Schneider, AISTATS, 2010]

  18. Learning Nonlinear Dynamic Models • An important issue • Linear model provides a severely restricted space of models • we know a model is wrong because the regression yields large residuals and low likelihoods • The nonlinear models are too powerful; they can fit anything! • Solution: restrict the space of nonlinear models • form the full kernel matrix • use a low-rank approximation of the kernel matrix

  19. Synthetic Nonlinear Data: Lorenz Attractor Estimated gradients by kernel UM

  20. Ordering by Temporal Smoothing

  21. Ordering by Temporal Smoothing

  22. Ordering by Temporal Smoothing

  23. Evaluation Criteria

  24. Results: 3D-1

  25. Results: 3D-2

  26. 3D-1: Algorithm Comparison

  27. 3D-2: Algorithm Comparison

  28. Methods for Real Data • Run k-means to cluster the data • Find an ordering of the cluster centers • TSP on pairwise L1 distances (TSP+L1) • OR • Temporal Smoothing Method (TSM) • Learn a dynamic model for the cluster centers • Initialize UM/PM with the learned model

  29. Gene Expression in Yeast Metabolic Cycle

  30. Gene Expression in Yeast Metabolic Cycle

  31. Results on Individual Genes

  32. Results over the whole space

  33. Cosine score in high dimensions Probability of random direction achieving a cosine score > 0.5 dimension

  34. Suppose we have some sequenced data linear dynamic model: perform a standard regression: what if the amount of data is not enough to regress reliably?

  35. Regularization for Regression add regularization to the regression: ridge regression: lasso: can the unsequenced data be used in regularization?

  36. Lyapunov Regularization Lyapunov equation relates dynamic model to steady state distribution: Q – covariance of steady state distribution • estimate Q from the unsequenced data! • optimize via gradient descent using the unpenalized or the ridge regression solution as the initial point

  37. Lyapunov Regularization: Toy Example -0.428 0.572 -1.043 -0.714 s = 1 A = • 2-d linear system • 2nd column of A fixed at the correct value • given 4 sequence points • given 20 unsequenced points

  38. Lyapunov Regularization: Toy Example

  39. Results on Synthetic Data Random 200 dimensional sparse (1/8) stable system

  40. Work in Progress … • cell cycle data from: [Zhou, Li, Yan, Wong, IEEE Trans on Inf Tech in Biomedicine, 2009] • 49 features on protein subcellular location • 34 sequences having a full cycle and length at least 30 were identified • another 11,556 are unsequenced • use the 34 sequences as ground truth and train on the unsequenced data A set of 100 sequenced images A tracking algorithm identified 34 sequences

  41. Preliminary Results: Protein Subcellular Location Dynamics normalized error cosine score

  42. Conclusions and Future Work • Demonstrated ability to learn (non)linear dynamic models from unsequenced data • Demonstrated method to use sequenced and unsequenced data together • Continuing efforts on real scientific data • Can we do this with hidden states?

  43. EXTRA SLIDES

  44. Real Data: Swinging Pendulum Video

  45. Results: Swinging Pendulum Video

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