1. 1 Relational Factor Graphs Lin Liao Joint work with Dieter Fox

2. 2 A Running Example Collective classification of a person’s significant places

3. 3 Features to Consider  Local features: Temporal: time of day, day of week, duration Geographic: near restaurants, near stores  Pair-wise features: Transitions: which place follows which place  Global features: Aggregates: number of homes or workplaces

4. 4 Which Graphical Model?  Option 1: Bayesian networks and Probabilistic Relational Models But the pair-wise relations may introduce cycles Place 1 Place 3 Place 4 Place 2

5. 5 Which Graphical Model?  Option 2: Markov networks and Relational Markov Networks But aggregations can introduce huge cliques and lose independence relations. Place 1 Place 3Place 4 Place 2 Number of homes

6. 6 Motivation  We want a relational probabilistic model that is Suitable to represent both undirected relations (e.g., pair-wise features) and directed relations (e.g., deterministic aggregation) Able to address some of the computational issues at the template level

7. 7 Outline  Representation Factor graphs [Kschischang et al. 2001, Frey 2003] Relational factor graphs  Inference Belief propagation Inference templates  Summation template based on FFT  Experiments

8. 8 Factor Graph  Undirected factor graph [Kschischang et al. 2001] Bipartite graph that includes both variable nodes (x 1,…,x N ) and factor nodes (f 1,…,f M ) Joint distribution of variables is proportional to the product of factor functions x1x1 x2x2 x3 x4 f1f1 f2f2 f3f3

9. 9 Factor Graph  Directed factor graph [Frey 2003] Allow some edges to be directed so as to unify Bayesian networks and Markov networks A valid graph should have no directed cycles x1x1 x2x2 f1f1 x3 x4 f3f3 f2f2

10. 10 Markov Network to Factor Graph Factors represent the potential functions Markov networkFactor graph

11. 11 Bayesian Network to Factor Graph Factors represent the conditional probability table Bayesian networkFactor graph

12. 12 Unify MN and BN + Local features Place labels Aggregation factor Number of homes Aggregate features

13. 13 Relational Factor Graph  A set of factor templates that can be used to instantiate (directed) factor graphs given data Representation template  Use SQL (similar to RMN)  Guarantee no directed cycles Inference template  Optimization within a factor (discussed later)

14. 14 Place Labeling: Schema

15. 15 Place Labeling: Transition Features Label1Label2Label3 Pair-wise factor

16. 16 Place Labeling: Aggregate Features Label1Label2Label3 + =Home? Bool variables Num of homes Aggregate feature

17. 17 Outline  Representation Factor graphs [Kschischang et al. 2001, Frey 2003] Relational factor graphs  Inference Belief propagation Inference templates  Summation template based on FFT  Experiments

18. 18 Inference in Factor Graph  Belief propagation: two types of messages Message from variable x to factor f Message from factor f to variable x n x : factors adjacent to x; n f : variables adjacent to f

19. 19 Inference Templates  Simplest case: specify the function f(n f ) and use the above formula to compute message f -> x Problem: complexity is exponential in the number of factor arguments. This can be very expensive for aggregation factors  Inference templates allow users to specify optimized algorithms at the template level Be in general form and easy to be shared Support template level complexity analysis

20. 20 Summation Templates + ….. x in 1 x in 2 x in 7 x in 8 x out

21. 21 Summation: Forward Message + ….. x in 1 x in 2 x in 7 x in 8 x out  Compute the distribution of the sum of independent variables x in 1, …., x in 8

22. 22 Summation: Forward Message  Convolution tree: each node can be computed using FFT; total complexity O(nlog 2 n)

23. 23 Summation: Backward Message + ….. x in 1 x in 2 x in 7 x in 8 x out  Message from x out defines a prior distribution of the sum. For each value of x in 2, compute the distribution of sum and weighted by the prior

24. 24 Summation: Backward Message  If we reuse the results cached for the forward message, complexity becomes O(nlogn)

25. 25 Summation Templates  By using convolution tree, FFT, and caching, the average complexity of passing a message through summation factor is O(nlogn), instead of exponential.

26. 26 Learning  Estimate the weights for probabilistic factors (local features, pair-wise features, and aggregate features)  Optimize the weights to maximize the conditional likelihood of the labeled training data The same algorithm as RMN

27. 27 Experiments  Two data sets: “Single” data set: one person’s GPS data for 4 months “Multiple” data set: one-week GPS data from 5 subjects  Six candidate labels: Home, Work, Shopping, Dining, Friend, Others  Get the geographic knowledge from Microsoft MapPoint Web Service

28. 28 How Much Aggregates Help Error rateMultipleSingle No aggregate28%9% With aggregate18%6%  Test on “multiple” data set: leave- one-subject-crossvalidation  Test on “single” data set: crossvalidation (train on 1 month, test on 3 months)

29. 29 How Efficient the Optimized BP

30. 30 Summary  Relational factor graph is SQL + (directed) factor graph  It is Suitable to represent both undirected relations and directed relations Convenient to use: no directed cycles Able to address computation issues at the template level