1. Modeling Annotated Data (SIGIR 2003) David M. Blei, Michael I. Jordan Univ. of California, Berkeley Presented by ChengXiang Zhai, July 10, 2003

2. A One-slide Summary  Problem: probabilistic modeling of annotated data (images + caption)  Method: Correspondence latent Dirichlet allocation (Corr- LDA) + two baseline models (GM-mixture, GM-LDA)  Evaluation  Held-out likelihood: Corr-LDA=GM-LDA > GM-mixture  Auto annotation: Corr-LDA>GM-mixture > GM-LDA  Image Retrieval: Corr-LDA>{GM-mixture, GM-LDA}  Conclusions: Corr-LDA is a good model

3. The Problem  Image/Caption Data: (r,w)  R={r1,…,rN}: Regions (primary data)  W={w1,…,wM}: Words (annotations)  R and W are different types  Need models for 3 tasks  Modeling join distribution: p(r,w) (for clustering)  Modeling conditional distr. P(w|r) (labeling an image, retrieval)  Modeling per-region word distr. P(w|ri) (labeling a region)

4. Three Generative Models  General  Region (ri) is modeled by a multivariate Gaussian with diagonal covariance  Word (wi) is modeled by a multinomial distribution  Assume k clusters  GM-Mixture: Gaussian-multinomial mixture  An image-caption pair belongs to exactly one cluster  Gaussian-multinomial LDA  An image-caption pair may belong to several clusters; each region/word belongs to exactly one cluster  Regions and words of the same image may belong to completely disjoint clusters  Correspondence LDA  An image-caption pair may belong to several clusters; each region/word belongs to exactly one cluster  A word must belong to exactly one of the clusters of the regions.

5. Detour, Probabilistic models for document/term Clustering…

6. A Mixture Model of Documents Select a group Generate a document (word sequence) C1C1 P(C 1 ) C2C2 P(C 2 ) P(w|C 1 ) P(w|C 2 ) CkCk … P(C k ) P(w|C k ) Maximum Likelihood Estimator

7. Applying EM Algorithm c1c1 c2c2 ckck d1d1 dndn Cluster/groupDocument Hidden variables z 11,…z 1k z ij  {0,1} z ij =1 iff d i is in cluster j z n1,…z nk Incomplete likelihood: Complete likelihood: E-step: compute E z |  old [L c (  |D)] M-step:  = argmax  E z |  old [L c (  |D)] Data: D={d1,…,dn} Compute p(z ij |d i,  old ) Compute expected counts for estimating 

8. EM Updating Formulas  Parameters:  =({p(C i )}, {p(w j |C i )})  Initialization: randomly set  0  Repeat until converge  E-step  M-step Practical issues: - “under-flow” - smoothing

9. Semi-supervised Classification  D=E(C 1 )  … E(C k )  U U – Unlabeled docs  Parameters:  =({p(C i )}, {p(w j |C i )})  Initialization: randomly set  0  Repeat until converge  E-step (only applied to d i in U)  M-step (pool real counts from E and expected counts from U) Smoothing +1 +|V| Essentially, set p(z ij )=1 for all d i in E(C j )!

10. End of Detour

11. GM-Mixture  Model:  Estimation: EM  Annotation:

12. Gaussian-Multinomial LDA  Model:  Estimation: Variational Bayesian  Annotation: Marginalization

13. Correspondence LDA  Model:  Estimation: Variational Bayesian  Annotation: Marginalization

14. Variational EM  General idea:  Using variational approximation to compute a lower bound of the likelihood in the E-step  Procedure:  Initialization  E-step: maximizing variational lower bound, usually involves iterations  M-step: Given variational distribution, estimate model parameters using ML

15. Experiment Results  Held-out likelihood: Corr-LDA=GM-LDA > GM- mixture  Auto annotation: Corr-LDA>GM-mixture > GM- LDA  Image Retrieval: Corr-LDA>{GM-mixture, GM- LDA}

16. Summary  A powerful generative model for annotated data (Corr-LDA)  Interesting empirical results