EdChang - Parallel Algorithms For Mining Large Scale Data

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EdChang - Parallel Algorithms For Mining Large Scale Data

  1. 1. Parallel Algorithms for Mining  g g Large‐ Large‐Scale Data Edward Chang Ed d Ch Director, Google Research, Beijing http://infolab.stanford.edu/ echang/ http://infolab stanford edu/~echang/ Ed Chang EMMDS 2009 1
  2. 2. Ed Chang EMMDS 2009 2
  3. 3. Ed Chang EMMDS 2009 3
  4. 4. Story of the Photos Story of the Photos • The photos on the first pages are statues of two “bookkeepers” displayed at the British Museum. One bookkeeper keeps a list of good people, and the other a list of bad. (Who is who, can you tell ? ☺) • When I first visited the museum in 1998, I did not take a photo of them to conserve films. During this trip (June 2009), capacity is no longer a concern or constraint. In fact, one can see kid grandmas all t ki photos, a l t of photos. T kids, d ll taking h t lot f h t Ten years apart, data volume explodes. • Data complexity also grows. • So, can these ancient “bookkeepers” still classify good from bad? Is their capacity scalable to the data dimension and data quantity ? Ed Chang EMMDS 2009 4
  5. 5. Ed Chang EMMDS 2009 5
  6. 6. Outline • Motivating Applications  – Q&A System – Social Ads • Key Subroutines – Frequent Itemset Mining [ACM RS 08] – Latent Dirichlet Allocation  [WWW 09, AAIM 09] L Di i hl All i – Clustering [ECML 08] – UserRank [Google TR 09] UserRank [Google TR 09] – Support Vector Machines [NIPS 07] • Distributed Computing Perspectives Distributed Computing Perspectives Ed Chang EMMDS 2009 6
  7. 7. Query: What are must‐see attractions at Yellowstone Ed Chang EMMDS 2009 7
  8. 8. Query: What are must‐see attractions at Yellowstone Ed Chang EMMDS 2009 8
  9. 9. Query: Must‐see attractions at Yosemite Ed Chang EMMDS 2009 9
  10. 10. Query: Must‐see attractions at Copenhagen Ed Chang EMMDS 2009 10
  11. 11. Query: Must‐see attractions at Beijing Hotel ads Ed Chang EMMDS 2009 11
  12. 12. Who is Yao Ming Ed Chang EMMDS 2009 12
  13. 13. Q&A Yao Ming Q&A Yao Ming Ed Chang EMMDS 2009 13
  14. 14. Who is Yao Ming Yao Ming Related Q&As Y Mi R l t d Q&A Ed Chang EMMDS 2009 14
  15. 15. Application: Google Q&A (Confucius) launched in China, Thailand, and Russia launched in China Thailand and Russia Differentiated Content DC U S C Search U Community CCS/Q&A Discussion/Question Trigger a discussion/question session during search Provide labels to a Q (semi-automatically) Given a Q, find similar Qs and their As (automatically) Evaluate quality of an answer, relevance and originality Evaluate E l t user credentials i a t i sensitive way d ti l in topic iti Route questions to experts Provide most relevant, high-quality content for Search to index Ed Chang EMMDS 2009 15
  16. 16. Q&A Uses Machine Learning Label suggestion using ML algorithms. • Real Time topic-to- topic (T2T) recommendation using ML algorithms. • Gi Gives out related hi h t l t d high quality links to previous questions before human answer appear. Ed Chang EMMDS 2009 16
  17. 17. Collaborative Filtering Books/Photos 1 1 1 1 1 1 1 1 1 Based on membership so far, 1 1 1 and memberships of others 1 1 1 1 1 Users 1 1 Predict further membership 1 1 1 1 1 1 1 1 1 1 1 1 1 Ed Chang EMMDS 2009 17
  18. 18. Collaborative Filtering Books/Photos Based on partially ? 1 1 1 ? observed matrix 1 ? 1 1 1 1 1 1 ? 1 1 1 1 ? 1 1 Predict unobserved entries 1 ? Users ? 1 1 U 1 1 ? I. Will user i enjoy photo j? 1 1 ? 1 ? 1 II. Will user i be interesting to user j? 1 ? 1 III. Will photo i be related to photo j? 1 1 1 1 1 ? Ed Chang EMMDS 2009 18
  19. 19. FIM based Recommendation FIM‐based Recommendation Ed Chang EMMDS 2009 19
  20. 20. FIM Preliminaries • Observation 1: If an item A is not frequent, any pattern contains  A won’t be frequent [R. Agrawal] q use a threshold to eliminate infrequent items  {A}   {A,B} • Observation 2: Patterns containing A are subsets of (or found  from) transactions containing A [J. Han] from) transactions containing A [J. Han] divide‐and‐conquer: select transactions containing A to form a  conditional database (CDB), and find patterns containing A from  that conditional database {A, B}, {A, C}, {A}   CDB A {A, B}, {B, C}  CDB B • Observation 3: To prevent the same pattern from being found in Observation 3: To prevent the same pattern from being found in  multiple CDBs, all itemsets are sorted by the same manner (e.g.,  by descending support) Ed Chang EMMDS 2009 20
  21. 21. Preprocessing • According to  f: 4 Observation 1, we  c: 4 count the support of  h f a: 3 facdgimp b: 3 fcamp each item by  m: 3 scanning the  abcflmo p: 3 fcabm database, and  database and eliminate those  bfhjo o: 2 fb infrequent items  d: 1 from the  from the bcksp e: 1 cbp transactions. g: 1 • According to  afcelpmn h: 1 fcamp i: 1 Observation 3, we  Observation 3, we k: 1 sort items in each  l:1 transaction by the  n: 1 order of descending g support value. Ed Chang EMMDS 2009 21
  22. 22. Parallel Projection • According to Observation 2, we construct CDB of item  A; then from this CDB, we find those patterns  ; , p containing A • How to construct the CDB of A?  – If t If a transaction contains A, this transaction should appear in  ti t i A thi t ti h ld i the CDB of A – Given a transaction {B, A, C}, it should appear in the CDB of  A, the CDB of B, and the CDB of C A the CDB of B and the CDB of C • Dedup solution: using the order of items: – sort {B,A,C} by the order of items  <A,B,C> – Put <> into the CDB of A – Put <A> into the CDB of B – Put <A,B> into the CDB of C Put <A B> into the CDB of C Ed Chang EMMDS 2009 22
  23. 23. Example of Projection fcamp p: { f c a m / f c a m / c b } fcabm m: { f c a / f c a / f c a b } fb b: { f c a / f / c } cbp a: { f c / f c / f c } fcamp c: { f / f / f } Example of Projection of a database into CDBs. Left: sorted transactions in order of f, c, a, b, m, p Right: conditional databases of frequent items Ed Chang EMMDS 2009 23
  24. 24. Example of Projection p j fcamp p: { f c a m / f c a m / c b } fcabm m: { f c a / f c a / f c a b } fb b: { f c a / f / c } cbp a: { f c / f c / f c } fcamp c: { f / f / f } Example of Projection of a database into CDBs. Left: sorted transactions; Right: conditional databases of frequent items Ed Chang EMMDS 2009 24
  25. 25. Example of Projection Example of Projection fcamp p: { f c a m / f c a m / c b } fcabm m: { f c a / f c a / f c a b } fb b: { f c a / f / c } cbp a: { f c / f c / f c } fcamp c: { f / f / f } Example of Projection of a database into CDBs. Left: sorted transactions; Right: conditional databases of frequent items Ed Chang EMMDS 2009 25
  26. 26. Recursive Projections [H. Li, et al. ACM RS 08] Recursive Projections [H Li et al ACM RS 08] MapReduce MapReduce MapReduce • Recursive projection form  Iteration 1 Iteration 2 Iteration 3 a search tree a search tree • Each node is a CDB b D|ab c D|abc • Using the order of items to  a D|a prevent duplicated CDBs. t d li t d CDB c D|ac • Each level of breath‐first  search of the tree can be  D c D|bc done by a MapReduce  done by a MapReduce b D|b iteration. • Once a CDB is small  enough to fit in memory,  enough to fit in memory c D|c we can invoke FP‐growth  to mine this CDB, and no  more growth of the sub more growth of the sub‐ tree. Ed Chang EMMDS 2009 26
  27. 27. Projection using MapReduce Map inputs Sorted transactions Map outputs Reduce inputs Reduce outputs (transactions) (with infrequent (conditional transactions) (conditional databases) (patterns and supports) key="": value items eliminated) key: value key: value key: value facdgimp fcamp p: m: fcam fca p:{fcam/fcam/cb} p:3, pc:3 p: { f c a m / f c a m / c b } p:3 pc:3 a: fc c: f mf:3 mc:3 abcflmo fcabm m: f c a b ma:3 b: f c a m: { f c a / f c a / f c a b } mfc:3 a: f c mfa:3 c: f mca:3 bfhjo fb b: f mfca:3 bcksp cbp p: c b b: {fca/f/c} b:3 afcelpmn fcamp b: c a: {fc/fc/fc} a:3 p: fcam af:3 m: fca ac:3 a: fc afc:3 c: f c:3 c: {f/f/f} cf:3 Ed Chang EMMDS 2009 27
  28. 28. Collaborative Filtering Indicators/Diseases 1 1 1 1 1 1 1 1 1 Based on membership so far, 1 1 1 als and memberships of others 1 1 1 1 Individua 1 1 1 Predict further membership 1 1 1 1 1 1 1 1 1 1 1 1 1 Ed Chang EMMDS 2009 28
  29. 29. Latent Semantic Analysis • Search Users/Music/Ads/Question – Construct a latent layer for better  for semantic matching ers sic/Ads/Answe • Example: – iPhone crack – Apple pie Users/Mus 1 recipe pastry for a 9 Documents inch double crust How to install apps on 9 apples, 2/1 cup, Apple mobile phones? brown sugar Topic Distribution • Other Collaborative Filtering Apps Other Collaborative Filtering Apps – Recommend Users  Users Topic Distribution – Recommend Music  Users – Recommend Ads  Users – Recommend Answers  Q R dA iPhone crack Apple pie User quries • Predict the ? In the light‐blue cells Ed Chang EMMDS 2009 29
  30. 30. Documents, Topics, Words Documents, Topics, Words • A document consists of a number of topics A document consists of a number of topics – A document is a probabilistic mixture of topics • Each topic generates a number of words Each topic generates a number of words – A topic is a distribution over words – The probability of the ith word in a document Ed Chang EMMDS 2009 30
  31. 31. Latent Dirichlet Allocation [M. Jordan 04] • α: uniform Dirichlet φ prior  α for per document d topic  for per document d topic distribution (corpus level  parameter) θ • β: uniform Dirichlet φ prior  f hl for per topic z word  distribution (corpus level  parameter) z • θd is the topic distribution of  doc d (document level) ( • zdj the topic if the jth word in  w φ β d, wdj the specific word (word  Nm K level) )  M Ed Chang EMMDS 2009 31
  32. 32. LDA Gibbs Sampling: Inputs And Outputs LDA Gibbs Sampling: Inputs And Outputs Inputs:  1. training data: documents as bags  words topics of words 2. parameter: the number of topics h b f docs docs Outputs:  1. by‐product: a co‐occurrence  1 by product: a co occurrence matrix of topics and documents. topics 2. model parameters: a co‐ words occurrence matrix of topics and  words. Ed Chang EMMDS 2009 32
  33. 33. Parallel Gibbs Sampling [aaim 09] Parallel Gibbs Sampling [aaim 09] Inputs:  1. training data: documents as bags  words topics of words 2. parameter: the number of topics h b f docs docs Outputs:  1. by‐product: a co‐occurrence  1 by product: a co occurrence matrix of topics and documents. topics 2. model parameters: a co‐ words occurrence matrix of topics and  words. Ed Chang EMMDS 2009 33
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  36. 36. Outline • Motivating Applications  – Q&A System – Social Ads • Key Subroutines – Frequent Itemset Mining [ACM RS 08] – Latent Dirichlet Allocation  [WWW 09, AAIM 09] L Di i hl All i – Clustering [ECML 08] – UserRank [Google TR 09] UserRank [Google TR 09] – Support Vector Machines [NIPS 07] • Distributed Computing Perspectives Distributed Computing Perspectives Ed Chang EMMDS 2009 36
  37. 37. Ed Chang EMMDS 2009 37
  38. 38. Ed Chang EMMDS 2009 38
  39. 39. Open Social APIs Open Social APIs 1 Profiles (who I am) 2 Stuff (what I have) Open Social Friends (who I know) 4 Activities (what I do) 3 Ed Chang EMMDS 2009 39
  40. 40. Activities Recommendations Applications Ed Chang EMMDS 2009 40
  41. 41. Ed Chang EMMDS 2009 41
  42. 42. Ed Chang EMMDS 2009 42
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  44. 44. Ed Chang EMMDS 2009 44
  45. 45. Task: Targeting Ads at SNS Users Task: Targeting Ads at SNS Users Users Ads Ed Chang EMMDS 2009 45
  46. 46. Mining Profiles, Friends & Activities  for Relevance for Relevance Ed Chang EMMDS 2009 46
  47. 47. Consider also User Influence • Advertisers consider  users who are h – Relevant – Influential • SNS Influence Analysis – Centrality – Credential – Activeness – etc. Ed Chang EMMDS 2009 47
  48. 48. Spectral Clustering [A. Ng, M. Jordan]] p g [ g, • Important subroutine in tasks of machine learning   and data mining – Exploit pairwise similarity of data instances – More effective than traditional methods e g k means More effective than traditional methods e.g., k‐means • Key steps – Construct pairwise similarity matrix Construct pairwise similarity matrix • e.g., using Geodisc distance – Compute the Laplacian matrix – A l i d Apply eigendecomposition iti – Perform k‐means Ed Chang EMMDS 2009 48
  49. 49. Scalability Problem Scalability Problem • Quadratic computation of nxn matrix • Approximation methods Dense Matrix Sparsification Nystrom Others t NN t-NN ξ-neighborhood ξ neighborhood … random greedy …. Ed Chang EMMDS 2009 49
  50. 50. Sparsification vs. Sampling p p g • Construct the dense  • Randomly sample l similarity matrix S similarity matrix S points, where l << n points where l << n • Sparsify S • Construct dense  • Compute Laplacian Compute Laplacian  similarity matrix [A B]  similarity matrix [A B] matrix L  between l and n points • Normalize A and B to be Normalize A and B to be  in Laplacian form • Apply ARPACLK on L R = A + A‐1/2BBTA‐1/2 ; R  A + A ;   • Use k‐means to cluster  R = U∑UT rows of V into k groups • k‐means Ed Chang EMMDS 2009 50
  51. 51. Empirical Study [song, et al., ecml 08] Empirical Study [song et al ecml 08] • Dataset: RCV1 (Reuters Corpus Volume I) – A filt d ll ti A filtered collection of 193,944 d f 193 944 documents in 103 t i 103 categories • Photo set: PicasaWeb – 637,137 photos • Experiments – Clustering quality vs. computational time • Measure the similarity between CAT and CLS  • Normalized Mutual Information (NMI) – Scalability Ed Chang EMMDS 2009 51
  52. 52. NMI Comparison (on RCV1) NMI Comparison (on RCV1) Nystrom method N t th d Sparse matrix approximation S ti i ti Ed Chang EMMDS 2009 52
  53. 53. Speedup Test on 637,137 Speedup Test on 637,137 Photos • K = 1000 clusters • A hi Achiever linear speedup when using 32 machines, after that,  li d h i 32 hi ft th t sub‐linear speedup because of increasing communication and  sync time Ed Chang EMMDS 2009 53
  54. 54. Sparsification vs. Sampling p p g Sparsification Nystrom, random  sampling Information Full n x n  Full n x n None similarity scores Pre‐processing  Pre processing O(n2) worst case; O(nl) l << n ) worst case;  O(nl), l << n Complexity  easily parallizable (bottleneck) Effectiveness Good Not bad (Jitendra M.,  PAMI) Ed Chang EMMDS 2009 54
  55. 55. Outline • Motivating Applications  – Q&A System – Social Ads • Key Subroutines – Frequent Itemset Mining [ACM RS 08] – Latent Dirichlet Allocation  [WWW 09, AAIM 09] L Di i hl All i – Clustering [ECML 08] – UserRank [Google TR 09] UserRank [Google TR 09] – Support Vector Machines [NIPS 07] • Distributed Computing Perspectives Distributed Computing Perspectives Ed Chang EMMDS 2009 55
  56. 56. Matrix Factorization Alternatives Matrix Factorization Alternatives exact approximate Ed Chang EMMDS 2009 56
  57. 57. PSVM [E. Chang, et al, NIPS 07] PSVM [E Chang et al NIPS 07] • Column‐based ICF Column based ICF – Slower than row‐based on single machine – Parallelizable on multiple machines Parallelizable on multiple machines • Changing IPM computation order to achieve  parallelization ll li ti Ed Chang EMMDS 2009 57
  58. 58. Speedup Ed Chang EMMDS 2009 58
  59. 59. Overheads Ed Chang EMMDS 2009 59
  60. 60. Comparison between Parallel  Computing Frameworks k MapReduce Project B MPI GFS/IO and task rescheduling Yes No No overhead between iterations +1 +1 Flexibility of computation model y p AllReduce only y AllReduce only y Flexible +0.5 +0.5 +1 Efficient AllReduce Yes Yes Yes +1 +1 +1 Recover from faults between Yes Yes Apps iterations +1 +1 Recover from faults within each Yes Yes Apps iteration +1 +1 Final Score for scalable machine 3.5 4.5 5 g learning Ed Chang EMMDS 2009 60
  61. 61. Concluding Remarks • Applications demand scalable solutions • Have parallelized key subroutines for mining massive data sets – Spectral Clustering [ECML 08] – Frequent Itemset Mining [ACM RS 08] – PLSA [KDD 08] – LDA [WWW 09] – UserRank  – Support Vector Machines [NIPS 07] • Relevant papers – http://infolab.stanford.edu/~echang/ • Open Source PSVM PLDA Open Source PSVM, PLDA – http://code.google.com/p/psvm/ – http://code.google.com/p/plda/ Ed Chang EMMDS 2009 61
  62. 62. Collaborators • Prof. Chih‐Jen Lin (NTU) ( ) • Hongjie Bai (Google) • Wen‐Yen Chen (UCSB) • Jon Chu (MIT) ( ) • Haoyuan Li (PKU) • Yangqiu Song (Tsinghua) Yangqiu Song (Tsinghua) • Matt Stanton (CMU) • Yi Wang (Google) g( g ) • Dong Zhang (Google) • Kaihua Zhu (Google) Ed Chang EMMDS 2009 62
  63. 63. References [1] Alexa internet. http://www.alexa.com/. [ ] [2] D. M. Blei and M. I. Jordan. Variational methods for the dirichlet process. In Proc. of the 21st international p conference on Machine learning, pages 373-380, 2004. [3] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent dirichlet allocation. Journal of Machine Learning Research, 3:993- 1022, 2003. [4] D. Cohn and H. Chang. Learning to probabilistically identify authoritative documents. In Proc. of the Seventeenth International Conference on Machine Learning, pages 167-174, 2000. [ ] [5] D. Cohn and T. Hofmann. The missing link - a probabilistic model of document content and hypertext connectivity. In g p yp y Advances in Neural Information Processing Systems 13, pages 430-436, 2001. [6] S. C. Deerwester, S. T. Dumais, T. K. Landauer, G. W. Furnas, and R. A. Harshman. Indexing by latent semantic analysis. Journal of the American Society of Information Science, 41(6):391-407, 1990. [7] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39(1):1-38, 1977. [8] S Ge a a d D. Ge a S oc as c relaxation, g bbs d s bu o s, a d the bayes a restoration o images. IEEE S. Geman and Geman. Stochastic e a a o , gibbs distributions, and e bayesian es o a o of ages Transactions on Pattern recognition and Machine Intelligence, 6:721-741, 1984. [9] T. Hofmann. Probabilistic latent semantic indexing. In Proc. of Uncertainty in Arti cial Intelligence, pages 289-296, 1999. [10] T. Hofmann. Latent semantic models for collaborative filtering. ACM Transactions on Information System, 22(1):89- 115, 2004. [11] A. McCallum, A. Corrada Emmanuel, and X. Wang. The author recipient topic model for topic and role discovery in Corrada-Emmanuel, author-recipient-topic social networks: Experiments with enron and academic email. Technical report, Computer Science, University of Massachusetts Amherst, 2004. [12] D. Newman, A. Asuncion, P. Smyth, and M. Welling. Distributed inference for latent dirichlet allocation. In Advances in Neural Information Processing Systems 20, 2007. [13] M. Ramoni, P. Sebastiani, and P. Cohen. Bayesian clustering by dynamics. Machine Learning, 47(1):91-121, 2002. Ed Chang EMMDS 2009 63
  64. 64. References (cont ) (cont.) [14] R. Salakhutdinov, A. Mnih, and G. Hinton. Restricted boltzmann machines for collaborative ltering. In Proc. Of the 24th international conference on Machine learning, p g 791-798, 2007. g g, pages , [15] E. Spertus, M. Sahami, and O. Buyukkokten. Evaluating similarity measures: a large-scale study in the orkut social network. In Proc. of the 11th ACM SIGKDD international conference on Knowledge discovery in data mining, pages 678-684, 2005. [16] M. Steyvers, P. Smyth, M. Rosen-Zvi, and T. Gri ths. Probabilistic author-topic models for information discovery. In Proc. of the 10th ACM SIGKDD international conference on Knowledge discovery and data mining, pages 306- 315, 2004. , [17] A. Strehl and J. Ghosh. Cluster ensembles - a knowledge reuse framework for combining multiple partitions. Journal on Machine Learning Research (JMLR), 3:583-617, 2002. [18] T. Zhang and V. S. Iyengar. Recommender systems using linear classi ers. Journal of Machine Learning Research, 2:313-334, 2002. [19] S. Zhong and J. Ghosh. Generative model-based clustering of documents: a comparative study. Knowledge and Information Sys e s ( o a o Systems (KAIS), 8 3 38 , 2005. S), 8:374-384, 005 [20] L. Admic and E. Adar. How to search a social network. 2004 [21] T.L. Griffiths and M. Steyvers. Finding scientific topics. Proceedings of the National Academy of Sciences, pages 5228-5235, 2004. [22] H. Kautz, B. Selman, and M. Shah. Referral Web: Combining social networks and collaborative filtering. Communitcations of the ACM, 3:63-65, 1997. [23] R. Agrawal, T. Imielnski, A. Swami. Mining association rules between sets of items in large databses. SIGMOD Rec., 22:207-116, 1993. [24] J. S. Breese, D. Heckerman, and C. Kadie. Empirical analysis of predictive algorithms for collaborative filtering. In Proceedings of the Fourteenth Conference on Uncertainty in Artifical Intelligence, 1998. [25] M.Deshpande and G. Karypis. Item-based top-n recommendation algorithms. ACM Trans. Inf. Syst., 22(1):143- 177, 2004. Ed Chang EMMDS 2009 64
  65. 65. References (cont ) (cont.) [26] B.M. Sarwar, G. Karypis, J.A. Konstan, and J. Reidl. Item-based collaborative filtering recommendation algorithms. In Proceedings of the 10th International World Wide Web Conference, pages 285-295, 2001. [27] M.Deshpande and G. Karypis. Item-based top-n recommendation algorithms. ACM Trans. Inf. Syst., 22(1):143- 177, 2004. [28] B.M. Sarwar, G. Karypis, J.A. Konstan, and J. Reidl. Item-based collaborative filtering recommendation algorithms. In Proceedings of the 10th International World Wide Web Conference, pages 285-295, 2001. [29] M. Brand. Fast online svd revisions for lightweight recommender systems. In Proceedings of the 3rd SIAM International Conference on Data Mining, 2003. [30] D. Goldbberg, D. Nichols, B. Oki and D. Terry. Using collaborative filtering to weave an information tapestry. Communication of ACM 35, 12:61-70, 1992. [31] P Resnik N Iacovou M Suchak P Bergstrom and J Riedl Grouplens: An open architecture for aollaborative P. Resnik, N. Iacovou, M. Suchak, P. Bergstrom, J. Riedl. filtering of netnews. In Proceedings of the ACM, Conference on Computer Supported Cooperative Work. Pages 175-186, 1994. [32] J. Konstan, et al. Grouplens: Applying collaborative filtering to usenet news. Communication ofACM 40, 3:77-87, 1997. [33] U. Shardanand and P. Maes. Social information filtering: Algorithms for automating “word of mouth”. In Proceedings of ACM CHI, 1:210-217, 1995. [34] G Ki d G. Kinden, B S ith and J Y k A B. Smith d J. York. Amazon.com recommendations: it d ti item-to-item collaborative filt i t it ll b ti filtering. IEEE I t Internet t Computing, 7:76-80, 2003. [35] T. Hofmann. Unsupervised learning by probabilistic latent semantic analysis. Machine Learning Journal 42, 1:177- 196, 2001. [36] T. Hofmann and J. Puzicha. Latent class models for collaborative filtering. In Proceedings of International Joint Conference in Artificial Intelligence, 1999. [ ] p [37] http://www.cs.carleton.edu/cs_comps/0607/recommend/recommender/collaborativefiltering.html p g [38] E. Y. Chang, et. al., Parallelizing Support Vector Machines on Distributed Machines, NIPS, 2007. [39] Wen-Yen Chen, Dong Zhang, and E. Y. Chang, Combinational Collaborative Filtering for personalized community recommendation, ACM KDD 2008. [40] Y. Sun, W.-Y. Chen, H. Bai, C.-j. Lin, and E. Y. Chang, Parallel Spectral Clustering, ECML 2008. Ed Chang EMMDS 2009 65

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