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Combinatorial Hodge Theory and Information Processing

Combinatorial Hodge Theory and Information Processing. 姚 远 2010.3. 25. Hodge Decomposition. Visual Image Patches. A nn Lee, Kim Pedersen, David Mumford (2003) studies statistical properties of 3x3 high contrast image patches of natural images (from Van Heteran’s database)

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Combinatorial Hodge Theory and Information Processing

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  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 MauriceKendall 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 • Peer-Review • Multicriteria

  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 = eT 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 v0 such that PT v0 = v0 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. 2002, http://cybermetrics.wlv.ac.uk/database/stats/data Link structure correlated with Research Ranking? 159 53 18 17 34 12 Web Link among Chinese Universities

  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 DATA: 2002,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 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: 叶阴宇 Stanford MS&E 林力行 (Lim Lek-Heng) UC Berkeley 姜筱晔 Stanford ICME Jiang-Lim-Yao-Ye, 2009, arxiv.org/abs/0811.1067, Math. Prog. Accepted. Many Others: Steve Smale, Gunnar Carlsson, Hang Li, Yizhou Wang, Art Owen, Persi Diaconis, Don Saari, et al.

  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 [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

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