1 / 27

Hilbert Space Embeddings of Conditional Distributions -- With Applications to Dynamical Systems

Hilbert Space Embeddings of Conditional Distributions -- With Applications to Dynamical Systems. Le Song Carnegie Mellon University Joint work with Jonathan Huang, Alex Smola and Kenji Fukumizu. Hilbert Space Embeddings. Summary statistics for distributions:. (mean). (covariance).

benitad
Download Presentation

Hilbert Space Embeddings of Conditional Distributions -- With Applications to Dynamical Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Hilbert Space Embeddings of Conditional Distributions--With Applications to Dynamical Systems Le Song Carnegie Mellon University Joint work with Jonathan Huang, Alex Smola and Kenji Fukumizu

  2. Hilbert Space Embeddings • Summary statistics for distributions: (mean) (covariance) (expected features)

  3. Hilbert Space Embeddings • Reconstruct from for certain kernels! Eg. RBF kernel . • Sample average converges to true mean at .

  4. Hilbert Space Embeddings Structured Prediction Embed Conditional Distributions p(y|x)  µ[p(y|x)] Dynamical Systems

  5. Embedding P(Y|X) • Simple case: X binary, Y vector valued • Need to embed • Solution: • Divide observations according to their • Average feature for each group What if X is continuous or structured?

  6. Embedding P(Y|X) • Goal: Estimate embedding from finite sample • Problem: Some X=x are never observed… Embed p(Y|X=x) for x never observed?

  7. Key Idea • If and are close, and will be similar. Generalize to unobserved X=x via a kernel on X

  8. Conditional Embedding Conditional Embedding Operator

  9. Conditional Embedding Operator generalization of covariance matrix

  10. Kernel Estimate • Empirical estimate converges at

  11. Sum and Product Rules Can do probabilistic inference with embeddings!

  12. Dynamical Systems Hidden State Maintain belief using Hilbert space embeddings observation

  13. Advantages • Applies to any domain with a kernel Eg. reals, rotations, permutations, strings • Applies to more general distributions Eg. multimodal, skewed

  14. Evaluation with Linear Dynamics • Linear dynamics model (Kalman filter optimal): Trajectory of Hidden States Observations

  15. -1 -2 x ~ h ~ N(0,10 I), N(0,10 I) Gaussian process noise, Gaussian observation noise 0.35 RKHS 0.3 RMS Error Kalman 0.25 0.2 2 4 6 Training Size ( Steps)

  16. -2 x ~ G h ~ (1,1/3), N(0,10 I) Skewed process noise, small observation noise 0.24 RKHS 0.22 RMS Error Kalman 0.2 2 4 6 Training Size ( Steps)

  17. -2 -2 -2 x ~ h ~ 0.5N(-0.2,10 I)+0.5N(0.2,10 I), N(0,10 I) Bimodal process noise, small observation noise 0.175 RKHS 0.17 Kalman 0.165 RMS Error 0.16 0.155 0.15 2 3 4 5 6 Training Size ( Steps)

  18. Camera Rotation Rendered using POV-ray renderer

  19. Predicted Camera Rotation Dynamical systems with Hilbert space embeddings works better

  20. Video from Predicted Rotation Video Rendered with Predicted Camera Rotation Original Video

  21. Conclusion • Embedding conditional distributions • Generalizes to unobserved X=x • Kernel estimate • Convergence of empirical estimate • Probabilistic inference • Application to dynamical systems learning and inference

  22. The End • Thanks

  23. Future Work • Estimate the hidden states during training? • Dynamical system = chain graphical model; how about tree and general graph topology?

  24. -2 -1 x ~ h ~ N(0,10 I), N(0,10 I) Small process noise, large observation noise 0.5 RKHS 0.4 RMS Error Kalman 0.3 2 4 6 n Training Size (2 Steps)

  25. Dynamics can Help 2.5 2 Trace Measure 1.5 1 -3 -2 -1 0 log10(Noise Variance)

  26. Dynamical Systems Hidden State Maintain belief using Hilbert space embeddings observation

  27. Conditional Embedding Operator • Desired properties: • Covariance operator: generalization of covariance matrix

More Related