1. Manifold Learning on Probabilistic Graphical Models 概率图上的流形学习 答辩人 : 邵元龙 导师 : 鲍虎军 教授 & 何晓飞 教授 浙江大学 CAD&CG 国家重点实验室 2010 年 3 月 5 日

2. Outline Background & Motivation Function Learning v.s. Statistical Modeling Manifold Regularized Variational Inference Algorithm Design & Examples In Depth Analysis Implementation Experimental Results

3. Function Learning Given data points, and a function space, find the optimal function, such that Regularization is Important!!

4. Statistical Modeling All quantities, no matter given or to be estimated, are random variables. Then we model the joint distribution.

5. e.g. Gaussian Mixture Model

6. Difficulties How many components are there? Should there be any “components” ?

7. Difficulties (continued) What if data reside on a non-trivial manifold

8. Efforts towards Non-Parametric, but …

9. What we want…

10. Review GMM Function Learning embedded.

11. Problem Formulation What to regularize? Where to regularize?

12. Manifold Learning

13. Manifold Assumption Y changes smoothly with X, and we have so should be small over manifold Minimizing it over the manifold,

14. Manifold Regularization

15. Transductive Learning

16. Problem Formulation What to regularize? Where to regularize?

17. Variational Inference For, define, a var. dist. Approximate the true posterior with it by minimizing the KL divergence

18. Manifold Regularized Variational Inference

19. How to Optimize?

20. Optimization Algorithm

21. An Illustration

22. Works Done Example Distribution Types Convergence Proof Convexity Analysis (More TODO) Computational Complexity Numerical Stability A Flexible Inference Engine

23. YASIE (Yet Another Statistical Inference Engine) Interface Design Inference Scheduling Type-Free Mixture Model Design Issues (e.g. Balance of Memory & Comp. Time)

24. Experiments Data Clustering Gaussian Mixture Model Image Annotation Link Mixture of Unigram

25. Image Annotation Model Link Mixture of Unigram

26. Image Similarity Graph “?” should be something like “Barcelona” ?

27. Image Annotation Performances

28. Image Annotation Examples

29. Any Question? 实验室的老师们：鲍虎军老师，何晓飞 老师，蔡登老师，刘新国老师，章国锋 老师，黄劲老师 …… 师兄师弟师妹们：董子龙，姜翰青，周 源，张驰原，林斌斌，薛维，瞿新泉， 姚冠红 …… 感谢你们一直以来给我的帮助！