1. Combinatorial Hodge Theory and Information Processing 姚 远 2010.3.25

2. Hodge Decomposition

3. Visual Image Patches Ann Lee, Kim Pedersen, David Mumford (2003) studies statistical properties of 3x3 high contrast image patches of natural images (from Van Heteran’s database) Gunnar Carlsson, Vin de Silva, Tigran Ishkhanov, Afra Zomorodian (2004-present) found those image patches concentrate around a 2-dimensional klein bottle imbedded in 7-sphere They build up simplicial complex from point cloud data 1-D Harmonic flows actually focus on densest region -- 3 major circles

4. 1-D Harmonic Flows on the space of 3x3 Image Patches 左上图： Klein Bottle of 3x3 Image Patch Space (Courtesy of Carlsson-Ishkhanov, 2007) 左下图： Harmonic flows focus on 3 major circles where most of data concentrate

5. Ranking Psychology: L. L. Thurstone (1928) (scaling) Statistics: M. Kendall (1930s, rank corellation), F. Mosteller, Bradley-Terry,…, P. Diaconis (group theory), etc. Economics: K. Arrow, A. Sen (voting and social choice theory, Nobel Laureates) Computer Science: Google’s PageRank, HITS, etc. We shall focus on ranking in the sequel as the construction of abstract simplicial complex is more natural and easier than point cloud data.

6. Had William Hodge met Maurice Kendall Hodge Theory Pairwise ranking The Bridge of Sighs in Cambridge, St John’s College

7. Ranking on Networks “Multicriteria” ranking/decision systems –Amazon or Netflix’s recommendation system (user-product) –Interest ranking in social networks (person-interest) –S&P index (time-price) –Voting (voter-candidate) “Peer-review” systems –Publication citation systems (paper-paper) –Google’s webpage ranking (web-web) –eBay’s reputation system (customer-customer)

8. Ranking on Networks Multicriteria mv1mv2mv3 usr11-- usr225- usr3--4 usr4325 Peer-Review

9. Characteristics Aforementioned ranking data are often –Incomplete: typically about 1% –Imbalanced: heterogeneously distributed votes –Cardinal: given in terms of scores or stochastic choices Pairwise ranking on graphs: implicitly or explicitly, ranking data may be viewed to live on a simple graph G=(V,E), where –V: set of alternatives (webpages, products, etc.) to be ranked –E: pairs of alternatives to be compared

10. Pagerank Model assumption: –A Markov chain random walk on networks, subject to the link structure Algorithm [Brin-Page’98] –Choose Link matrix L, where L(i,j)=# links from i to j. –Markov matrix M=D -1 L, where D = e T L, e is the all-one vector. –Random Surfer model: E is all-one matrix –PageRank model: P = c M + (1-c) E/n, where c = 0.85 chosen by Google. –Pagerank vector: the primary eigenvector v 0 such that P T v 0 = v 0 Note: SVD decomposition of L gives HITS [Kleinberg’99] algorithm. Problem: Can we drop Markov Chain model assumption?

11. Netflix Customer-Product Rating

12. Netflix example continued The first order statistics, mean score for each product, is often inadequate because of the following: –most customers would rate just a very small portion of the products –different products might have different raters, whence mean scores involve noise due to arbitrary individual rating scales (right figure) How about high order statistics?

13. From 1st order to 2nd order: Pairwise Ranking

14. Pairwise Ranking Continued

15. Another View on Pagerank Define pairwise ranking: Where P is the Pagerank Markov matrix. Claim: if P is a reversible Markov chain, i.e. Then

16. Skew-Symmetric Matrices of Pairwise Ranking

17. Pairwise Ranking of Top 10 IMDB Movies Pairwise ranking graph flow among top 10 IMDB movies

18. Web Link among Chinese Universities 2002, http://cybermetrics.wlv.ac.uk/database/stats/datahttp://cybermetrics.wlv.ac.uk/database/stats/data Link structure correlated with Research Ranking ? 159 53 34 18 17 12

19. Classical Ordinal Rank Aggregation Problem

20. Rank Aggregation Problem

21. Answer: Not Always!

22. Triangular Transitivity

23. Hodge Decomposition: Matrix Theoretic

24. Hodge Decomposition: Graph Theoretic

25. Hodge Theory: Geometric Analysis on Graphs (and Complexes) 图 ( 复形 ) 上的几何分析

26. Clique Complex of a Graph

27. Discrete Differential Forms

28. Discrete Exterior Derivatives: coboundary maps

29. Curl ( 旋度 ) and Divergence ( 散度 )

30. Basic Alg. Top.: Boundary of Boundary is Empty

31. High Dim. Combinatorial Laplacians

32. Hodge Decomposition Theorem

33. Hodge Decomposition Theorem: Illustration

34. Harmonic rankings: locally consistent but globally inconsistent

35. Rank Aggregation as Projection

36. Don Saari ’ s Geometric Illustration of Different Projections

37. Measuring Inconsistency by Curls Define the cyclicity ratio by This measures the total inconsistency within the data and model w.

38. Choose 6 Netflix Movies with Dynamic Drifts

39. Model Selection by Cyclicity Ratio MRQE: Movie-Review-Query-Engine (http://www.mrqe.com/)

40. Just for fun: Chinese Universities (mainland) Ranking RankResearch 2002 PagerankHITS Authority HITS Hub Pku.edu.cn 1221 Tsinghua.ed u.cn 1116 Fudan.edu. cn 375021 DATA: 2002, http://cybermetrics.wlv.ac.uk/database/stats/data http://cybermetrics.wlv.ac.uk/database/stats/data

41. Further Application I: Scanpath via Eyetracker with Yizhou Wang and Wei Wang (PKU) et al.

42. Further Application II: Click-Pattern Analysis queryIDSearch ResultsClick Pattern Frequency q1[["u1","u2","u3"],["u4","u5","u6"]][[],[2]]1 q1[["u1","u2","u3"],["u7","u6","u8"]][[2],[]]1 q1[["u1","u2","u9"],["u10","u11","u7","u8","u4"]][[],[4]]1 q1[["u12","u2","u9","u3","u13","u14"]][[5]]1 q1[["u12","u2","u9"]][[1]]1 Pairwise Comparison discloses that ‘url1’ is less preferred to ‘url13’, in a contrast to the frequency count. with Hang LI (MSR-A) et al.

43. Acknowledgement Collaborators: 林力行 (Lim Lek-Heng) UC Berkeley 姜筱晔 Stanford ICME 叶阴宇 Stanford MS&E Many Others : Steve Smale, Gunnar Carlsson, Hang Li, Yizhou Wang, Art Owen, Persi Diaconis, Don Saari, et al. Jiang-Lim-Yao-Ye, 2009, arxiv.org/abs/0811.1067, Math. Prog. Accepted.

44. Reference [Friedman1996] J. Friedman, Computing Betti Numbers via Combinatorial Laplacians. Proc. 28th ACM Symposium on Theory of Computing, 1996. http://www.math.ubc.ca/~jf/pubs/web_stuff/newbetti.html http://www.math.ubc.ca/~jf/pubs/web_stuff/newbetti.html [Forman] R. Forman, Combinatorial Differential Topology and Geometry, New perspective in Algebraic Combinatorics, MSRI publications, http://math.rice.edu/~forman/combdiff.ps [David1988] David, H.A. (1988). The Method of Paired Comparisons. New York: Oxford University Press. [Saari1995] D. Saari, Basic Geometry of Voting, Springer, 1995 44