1 / 37

Learning Inhomogeneous Gibbs Models

Learning Inhomogeneous Gibbs Models. Ce Liu celiu@microsoft.com. How to Describe the Virtual World. Histogram. Histogram: marginal distribution of image variances Non Gaussian distributed. Texture Synthesis (Heeger et al, 95). Image decomposition by steerable filters Histogram matching.

aure
Download Presentation

Learning Inhomogeneous Gibbs Models

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. Learning Inhomogeneous Gibbs Models Ce Liu celiu@microsoft.com

  2. How to Describe the Virtual World

  3. Histogram • Histogram: marginal distribution of image variances • Non Gaussian distributed

  4. Texture Synthesis (Heeger et al, 95) • Image decomposition by steerable filters • Histogram matching

  5. FRAME (Zhu et al, 97) • Homogeneous Markov random field (MRF) • Minimax entropy principle to learn homogeneous Gibbs distribution • Gibbs sampling and feature selection

  6. Our Problem • To learn the distribution of structural signals • Challenges • How to learn non-Gaussian distributions in high dimensions with small observations? • How to capture the sophisticated properties of the distribution? • How to optimize parameters with global convergence?

  7. Inhomogeneous Gibbs Models (IGM) A framework to learn arbitrary high-dimensional distributions • 1D histograms on linear features to describe high-dimensional distribution • Maximum Entropy Principle– Gibbs distribution • Minimum Entropy Principle– Feature Pursuit • Markov chain Monte Carlo in parameter optimization • Kullback-Leibler Feature (KLF)

  8. 1D Observation: Histograms • Feature f(x): Rd→ R • Linear feature f(x)=fTx • Kernel distance f(x)=||f-x|| • Marginal distribution • Histogram

  9. Intuition

  10. Learning Descriptive Models =

  11. Learning Descriptive Models • Sufficient features can make the learnt model f(x) converge to the underlying distribution p(x) • Linear features and histograms are robust compared with other high-order statistics • Descriptive models

  12. Maximum Entropy Principle • Maximum Entropy Model • To generalize the statistical properties in the observed • To make the learnt model present information no more than what is available • Mathematical formulation

  13. Intuition of Maximum Entropy Principle

  14. Inhomogeneous Gibbs Distribution • Solution form of maximum entropy model • Parameter: Gibbs potential

  15. Estimating Potential Function • Distribution form • Normalization • Maximizing Likelihood Estimation (MLE) • 1st and 2nd order derivatives

  16. Parameter Learning • Monte Carlo integration • Algorithm

  17. y x Gibbs Sampling

  18. Minimum Entropy Principle • Minimum entropy principle • To make the learnt distribution close to the observed • Feature selection

  19. Feature Pursuit • A greedy procedure to learn the feature set • Reference model • Approximate information gain

  20. Proposition The approximate information gain for a new feature is and the optimal energy function for this feature is

  21. Kullback-Leibler Feature • Kullback-Leibler Feature • Pursue feature by • Hybrid Monte Carlo • Sequential 1D optimization • Feature selection

  22. Acceleration by Importance Sampling • Gibbs sampling is too slow… • Importance sampling by the reference model

  23. Flowchart of IGM Obs Samples Obs Histograms IGM MCMC Syn Samples Feature Pursuit KL Feature KL<e N Y Output

  24. Toy Problems (1) Feature pursuit Synthesized samples Gibbs potential Observed histograms Synthesized histograms Circle Mixture of two Gaussians

  25. Toy Problems (2) Swiss Roll

  26. Applied to High Dimensions • In high-dimensional space • Too many features to constrain every dimension • MCMC sampling is extremely slow • Solution: dimension reduction by PCA • Application: learning face prior model • 83 landmarks defined to represent face (166d) • 524 samples

  27. Face Prior Learning (1) Observed face examples Synthesized face samples without any features

  28. Face Prior Learning (2) Synthesized with 10 features Synthesized with 20 features

  29. Face Prior Learning (3) Synthesized with 30 features Synthesized with 50 features

  30. Observed Histograms

  31. Synthesized Histograms

  32. Gibbs Potential Functions

  33. Learning Caricature Exaggeration

  34. Synthesis Results

  35. Learning 2D Gibbs Process

  36. CSAIL Thank you! celiu@csail.mit.edu

More Related