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Extrapolating PageRank Computations for Faster Convergence
- 1. Extrapolation Methods for Accelerating PageRank Computations Sepandar D. Kamvar Taher H. Haveliwala Christopher D. Manning Gene H. Golub Stanford University
- 4. Link Counts Linked by 2 Important Pages Linked by 2 Unimportant pages Sep’s Home Page Taher’s Home Page Yahoo! CNN DB Pub Server CS361
- 6. Definition of PageRank Yahoo! CNN DB Pub Server Taher Sep 1/2 1/2 1 1 0.1 0.1 0.1 0.05 0.25
- 7. PageRank Diagram Initialize all nodes to rank 0.333 0.333 0.333
- 8. PageRank Diagram Propagate ranks across links (multiplying by link weights) 0.167 0.167 0.333 0.333
- 9. PageRank Diagram 0.333 0.5 0.167
- 10. PageRank Diagram 0.167 0.167 0.5 0.167
- 11. PageRank Diagram 0.5 0.333 0.167
- 12. PageRank Diagram After a while… 0.4 0.4 0.2
- 14. Matrix Notation 0 .2 0 .3 0 0 .1 .4 0 .1 = .1 .3 .2 .3 .1 .1 .2 . 1 .3 .2 .3 .1 .1
- 15. Matrix Notation Find x that satisfies: . 1 .3 .2 .3 .1 .1 0 .2 0 .3 0 0 .1 .4 0 .1 = .1 .3 .2 .3 .1 .1 .2
- 20. Power Method u 1 1 u 2 2 u 3 3 u 4 4 u 5 5 Express x (0) in terms of eigenvectors of A
- 21. Power Method u 1 1 u 2 2 2 u 3 3 3 u 4 4 4 u 5 5 5
- 22. Power Method u 1 1 u 2 2 2 2 u 3 3 3 2 u 4 4 4 2 u 5 5 5 2
- 23. Power Method u 1 1 u 2 2 2 k u 3 3 3 k u 4 4 4 k u 5 5 5 k
- 24. Power Method u 1 1 u 2 u 3 u 4 u 5
- 27. Power Method u 1 1 u 2 2 u 3 3 u 4 4 u 5 5 u 1 1 u 2 2 2 u 3 3 3 u 4 4 4 u 5 5 5 u 1 1 u 2 2 2 2 u 3 3 3 2 u 4 4 4 2 u 5 5 5 2
- 29. Our Approach u 1 u 2 u 3 u 4 u 5 Estimate components of current iterate in the directions of second two eigenvectors, and eliminate them.
- 31. Using Successive Iterates u 1 x (0) u 1 u 2 u 3 u 4 u 5
- 32. Using Successive Iterates u 1 x (1) x (0) u 1 u 2 u 3 u 4 u 5
- 33. Using Successive Iterates u 1 x (1) x (0) x (2) u 1 u 2 u 3 u 4 u 5
- 34. Using Successive Iterates x (0) u 1 x (1) x (2) u 1 u 2 u 3 u 4 u 5
- 35. Using Successive Iterates x (0) x’ = u 1 x (1) u 1 u 2 u 3 u 4 u 5
- 42. Results Quadratic Extrapolation speeds up convergence. Extrapolation was only used 5 times!
- 43. Results Extrapolation dramatically speeds up convergence, for high values of c (c=.99)
Notas del editor
- Why does the Power Method Work?
- Assume that lambda 1 is less than 1 and all other eigenvalues are strictly less than 1.
- Here, talk about in the past, how lambda 2 is often close to 1, so the power method is not useful. However, in our case,
- Note : derivation given here is slightly different from what’s in the paper the one here is perhaps more intuitive the one in the paper is more compact