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Fast matrix primitives for ranking, link-prediction and more
- 1. Fast matrix primitives for ranking, communities ! and more. David F. Gleich! David Gleich · Purdue Netflix 1 Computer Science! Purdue University!
- 2. Netflix 2 David Gleich · Purdue
- 3. error 1 Models Previous work – and algorithms for high performance ! from the PI tackled net- computations matrix and network FIGURE 6 std 2 work alignment with matrix methods =for cm edge Std, s 0.39 Big data(b)methods overlap: SIMAX ‘09, SISC ‘11,MapReduce ‘11, ICASSP ’12 1 j i i0 Overlap Overlap j0 error SC ‘05, WAW ‘07, SISC ‘10, WWW ’10, … Massive matrix " computations std 0 0 Fast & Scalable" Network centrality 0 20 10 10 A L B Tensor eigenvalues" 0 (d) Std, s = 1.95 cm Ax = b min kAx bk Ax = x This proposal is for matchand a power method Network alignment tensor ing triangles using P methods: on multi-threaded maximize Tijk xi xj xk model compared to the prediction standard debble locations at the final time for two values of ICDM ‘09, SC ‘11, TKDE ‘13 = 1.95 cm. (Colors are visible in the electronic ijk n and kxk2 = 1 subject to distributed j Triangle j X i s i approximately twenty minutes to construct using (next) architectures k [x ]i = ⇢ · ( Tijk xj xk + xi ) k s. jk s- involved a few pre- and post-processing steps: ta where ! ensures the 2-norm m Aria, globally transpose the data, compute the g errors. The preprocessing steps took approx- SSHOPM method due to " nd 0 0 Data clustering WSDM ‘12, KDD ‘12, CIKM ’13 … 0 A recise timing information, L we do notB but report David Kolda and Mayo Gleich · Purdue Netflix 3 t r o s. g n. o
- 4. The talk ends, you believe -- whatever you want to. Image from rockysprings, deviantart, CC share-alike 4 Everything in the world can be explained by a matrix, and we see how deep the rabbit hole goes
- 5. Matrix computations in a red-pill David Gleich · Purdue Netflix 5 Solve a problem better by exploiting its structure!
- 6. Problem 1 – (Faster) ! Recommendation as link prediction WHY NO PREPROCESSING? Top-k predicted “links” are movies to watch! David F. Gleich (Purdue) David Gleich · Purdue Emory Math/CS Seminar Netflix 6 Pairwise scores give user similarity 19 of 47
- 7. David Gleich · Purdue Netflix 7 Problem 2 – (Better) ! Best movies
- 8. Matrix computations in a red-pill David Gleich · Purdue Netflix 8 Solve a problem better by exploiting its structure!
- 9. Matrix structure Netflix graph Movies " “liked” (>3 stars?) Problem 1! Adjacency matrix Normalized Laplacian matrix Random walk matrix Netflix matrix 1 1 4 5 5 Problem 2" Pairwise comparison matrix David Gleich · Purdue Netflix 9 5
- 10. Problem 1 – (Faster) ! Recommendation as link prediction WHY NO PREPROCESSING? Top-k predicted “links” are movies to watch! David F. Gleich (Purdue) David Gleich · Purdue Emory Math/CS Seminar Netflix 10 Pairwise scores give user similarity 19 of 47
- 11. z score (edge-based) is movie = X ↵ ✓ pred. on ` `=1 1 X num. paths of length ` from user to movie ✓ ◆ user k= (↵ A ) ind. vec. `=1 {z } | ` Math/CS Seminar Emory ` ⌘ei ◆ David Gleich · Purdue Netflix 11 1 Movie prediction" vector ) Matrix based link predictors
- 12. Matrix based link predictors 1 X ✓ ◆ user k= (↵ A ) ind. vec. `=1 {z } | ` ` ⌘ei Neumann Carl Neumann 1 X k =0 ↵A)k = ei (tA)k David Gleich · Purdue Netflix 12 (I
- 13. The Katz score (edge-based) is Matrix based link predictors (I ↵A)k = ei PageRank (I ↵P)x = ei Semi-super." learning Heat kernel (IDavid F. Gleich (Purdue) = ei ↵L)x exp{↵P}x = ei Emory Math/CS Seminar They all look at sums of " damped paths, but " change the details, slightly David Gleich · Purdue Netflix 13 Katz
- 14. Matrix based link predictors are localized! PageRank scores for one node! Crawl of flickr from 2006 ~800k nodes, 6M edges, alpha=1/2 0 1.5 error ||xtrue – xnnz||1 10 1 0.5 −10 10 −15 0 2 4 plot(x) 6 8 10 5 x 10 10 0 10 2 4 10 6 10 10 nonzeros David Gleich · Purdue Netflix 14 0 −5 10
- 15. Matrix based link predictors are localized! KATZ SCORES ARE LOCALIZED David F. Gleich (Purdue) Emory Math/CS Seminar 32 of 47 David Gleich · Purdue Netflix 15 Up to 50 neighbors is 99.65% of the total mass
- 16. Matrix computations in a red-pill David Gleich · Purdue Netflix 16 Solve a problem better by exploiting its structure!
- 17. How do we compute them fast? PageRank xj = ↵ X i neigh. of j xi deg(i) + 1 if j is the target user w/ access to in-links & degs. w/ access to out-links PageRankPull PageRankPush xj(k+1) (k) ↵xa /6 (k) ↵xb /2 (k) ↵xc /3 = fj xj(k+1) ↵ X i!j xi(k ) /degi = fj Let b a c j = blue node (k+1) = xj(k) + rj (k +1) =0 then xj Update r(k +1) rj (k (k) ra +1) = ra + ↵rj(k ) /3 (k (k) rb +1) = rb + ↵rj(k ) /3 (k (k) rc +1) = rc + ↵rj(k ) /3 David Gleich · Purdue Netflix 17 (k +1) Solve for xj j = blue node
- 18. We have good theory for this algorithm … David Gleich · Purdue Netflix 18 … and even better empirical performance.
- 19. Theory Andersen, Chung, Lang (2006)! For PageRank, “fast runtimes” and “localization” Bonchi, Esfandiar, Gleich, et al. (2010/2013)! For Katz, “fast runtimes” David Gleich · Purdue Netflix 19 Kloster, Gleich (2013)! For Katz, Heat Kernel, " “fast runtimes” and “localization”" (assuming power-law degrees)
- 20. Accuracy vs. work ! (Heat kernel) dblp−cc dblp collaboration graph, 225k vertices 1 0.6 tol=10−5 tol=10−4 0.4 @10 @25 0.2 @100 @1000 0 −2 −1 0 10 10 10 Effective matrix−vector products David Gleich · Purdue Netflix 20 Precision 0.8 For the dblp collaboration graph, we study the precision in finding the 100 largest nodes as we vary the work. This set of 100 does not include the nodes immediate neighbors. (One column, but representative)
- 21. David Gleich · Purdue Netflix 21 Empirical runtime (Katz) TIMING
- 22. 40 Never got to try it … analytics test on HelloMovies.com Need to ix now matr Netflix 60 80 Ran out of money once we had the algorithms … promising initial results though! I collaborate with the company behind He David Gleich · Purdue Netflix 22 Note 1 1 1 1 1 1 1 1
- 23. David Gleich · Purdue Netflix 23 Problem 2 – (Better) ! Best movies
- 24. Which is a better list of good DVDs? Lord of the Rings 3: The Return of … Lord of the Rings 3: The Return of … Lord of the Rings 1: The Fellowship Lord of the Rings 1: The Fellowship Lord of the Rings 2: The Two Towers Lord of the Rings 2: The Two Towers Lost: Season 1 Star Wars V: Empire Strikes Back Battlestar Galactica: Season 1 Raiders of the Lost Ark Fullmetal Alchemist Star Wars IV: A New Hope Trailer Park Boys: Season 4 Shawshank Redemption Trailer Park Boys: Season 3 Star Wars VI: Return of the Jedi Tenchi Muyo! Lord of the Rings 3: Bonus DVD Shawshank Redemption The Godfather Nuclear Norm " based rank aggregation (the mean rating) (not matrix completion on the netflix rating matrix) David Gleich · Purdue Netflix 24/40 Standard " rank aggregation"
- 25. Rank Aggregation Given partial orders on subsets of items, rank aggregation is the problem of finding an overall ordering. Voting Find the winning candidate Program committees Find the best papers given reviews Dining Find the best restaurant in Chicago David Gleich · Purdue Netflix 25/40
- 26. Ranking is really hard John Kemeny Ken Arrow All rank aggregations involve some measure of compromise A good ranking is the “average” ranking under a permutation distance NP hard to compute Kemeny’s ranking David Gleich · Purdue Netflix 26/40 Dwork, Kumar, Naor, ! Sivikumar
- 27. Supposewe had scores Suppose we had scores Let be the score of the ith movie/song/paper/team to rank Suppose we can compare the ith to jth: is skew-symmetric, rank 2. Also works for with an extra log. Numerical ranking is intimately intertwined with skew-symmetric matrices Kemeny and Snell, Mathematical Models in Social Sciences (1978) David F. Gleich (Purdue) KDD 2011 David Gleich · Purdue Netflix 6/20 27/40 Then
- 28. Using ratings as comparisons Arithmetic Mean Ratings induce various skewsymmetric matrices. From David 1988 – The Method of Paired Comparisons David Gleich · Purdue Netflix 28/40 Log-odds
- 29. Extracting the scores Extracting the scores do we have? Do we trust all Not really. David F. Gleich (Purdue) 105 101 101 105 Number of Comparisons ? Netflix data 17k movies, 500k users, 100M ratings– 99.17% filled KDD 2011 David Gleich · Purdue Netflix 29/40 How many Most. 107 Movie Pairs Given with all entries, then is the Borda count, the least-squares solution to 8/20
- 30. Only partial info? COMPLETE IT! Only partial info? Complete it! Let be known for We trust these scores. Goal Find the simplest skew-symmetric matrix that matches the data noiseless Both of these are NP-hard too. David F. Gleich (Purdue) David Gleich · Purdue KDD 2011 Netflix 30/40 noisy 9/20
- 31. From a French nuclear test in 1970, imageNetflix from http://picdit.wordpress.com/2008/07/21/8David Gleich · Purdue insane-nuclear-explosions/ 31/40 Solution GO NUCLEAR!
- 32. The ranking algorithm The Ranking Algorithm 0. INPUT (ratings data) and c (for trust on comparisons) 1. Compute from 2. Discard entries with fewer than c comparisons 3. Set to be indices and values of what’s left 4. = SVP( ) David Gleich · Purdue Netflix 32/40 5. OUTPUT
- 33. Exact recovery Exactrecovery results Fraction of trials recovered indices. Instead we view the following theorem as providing intuition for the noisy problem. Consider the operator basis for Hermitian matrices: H = S [ K [ D where p S = {1/ 2(ei eT + ej eT ) : 1 i < j n}; j i David Gross showed how to recover Hermitian matrices. p K = {ı/ 2(ei eT ej eT ) : 1we get n}; exact j i i.e. the conditions under which i < j the Note that D = {ei eT : 1 i n}. i 1 0.8 0.6 0.4 0.2 0 2 10 is Hermitian. Thus our new result! T Theorem 5. Let s be centered, i.e., s e = 0. Let Y = seT esT where ✓ = maxi s2 /(sT s) and ⇢ = ((maxi si ) i (mini si ))/ksk. Also, let ⌦ ⇢ H be a random set of elements with size |⌦| O(2n⌫(1 + )(log n)2 ) where ⌫ = max((n✓ + 1)/4, n⇢2 ). Then the solution of minimize kXk⇤ Figure ity of about both th §6.1 fo 6.1 R The fi subject to trace(X W i ) = trace((ıY ) W i ), W i 2 ⌦ ability o the nois is equal to ıY with probability at least 1 n . with un These a The proof of this theorem follows directly by Theorem 4 if Netflix = se David Gleich · Purdue Y ⇤ 33/40 ⇤
- 34. Recovery Discussion and Experiments Confession If , then just look at differences from a connected set. Constants? Not very good. Intuition for the truth. David Gleich · Purdue Netflix 34
- 35. Recovery Discussion and Experiments Recovery Experiments look at differences from Confession If , then just a connected set. Constants? Not very good. Intuition for the truth. KDD 2011 16/20 David Gleich · Purdue Netflix 35/40 David F. Gleich (Purdue)
- 36. Evaluation Nuclear norm ranking Mean rating 1 Median Kendall’s Tau 0.9 0.8 20 10 5 2 1.5 0.7 0.6 0.5 0.9 0.8 0.7 0.6 0.5 0 0.2 0.4 0.6 Error 0.8 1 0 0.2 0.4 0.6 Error 0.8 1 Figure 3: The performanceDavid Gleich · Purdue of our algorithm Netflix (left) 36/40 Median Kendall’s Tau 1
- 37. Tie in with PageRank Another way to compute the scores is through a close relative of PageRank and the linkprediction methods. Massey or Colley methods (2I + D A)s = “differeneces” (L + 2D 1 )x = “scaled differences” David Gleich · Purdue Netflix 37/40
- 38. Ongoing Work Finding communities in large networks ! We have the best community finder (as of CIKM2013)" Whang, Gleich, Dhillon (CIKM) Fast clique detection! We have the fastest solver for max-clique problems, useful for computing temporal strong components (Rossi, Gleich, et al. arXiv) Scalable network alignment ! w v s Overlap r & Low-rank clustering with features + links! wtu u t A L B & Evolving network analysis! David Gleich · Purdue Netflix 38 & Scalable, distributed implementations ! of fast graph kernels!
- 39. References ! Papers Gleich & Lim, KDD 2011 – Nuclear Norm Ranking" Esfandiar, Gleich, Bonchi et al. – WAW2010, J. Internet. Math. 2013" Kloster & Gleich, WAW2013, arXiv 1310.3423 Code! www.cs.purdue.edu/homes/dgleich/codes! bit.ly/dgleich-code Supported by NSF CAREER 1149756-CCF www.cs.purdue.edu/homes/dgleich David Gleich · Purdue Netflix 39 ! !