1. 1 Algorithms for Large Data Sets Ziv Bar-Yossef Lecture 5 April 23, 2006 http://www.ee.technion.ac.il/courses/049011

2. 2 Ranking Algorithms

3. 3 PageRank [Page, Brin, Motwani, Winograd 1998] Motivating principles  Rank of p should be proportional to the rank of the pages that point to p Recommendations from Bill Gates & Steve Jobs vs. from Moishale and Ahuva  Rank of p should depend on the number of pages “co-cited” with p Compare: Bill Gates recommends only me vs. Bill Gates recommends everyone on earth

4. 4 Then:  r is a non-negative normalized left eigenvector of B with eigenvalue 1 PageRank, Attempt #1 Additional Conditions:  r is non-negative:r ≥ 0  r is normalized:||r|| 1 = 1 B = normalized adjacency matrix:

5. 5 PageRank, Attempt #1 Solution exists only if B has eigenvalue 1 Problem: B may not have 1 as an eigenvalue  Because some of its rows are 0.  Example:

6. 6 = normalization constant  r is a non-negative normalized left eigenvector of B with eigenvalue 1/ PageRank, Attempt #2

7. 7 Any nonzero eigenvalue  of B may give a solution  = 1/   r = any non-negative normalized left eigenvector of B with eigenvalue  Which solution to pick?  Pick a “principal eigenvector” (i.e., corresponding to maximal  ) How to find a solution?  Power iterations PageRank, Attempt #2

8. 8 Problem #1: Maximal eigenvalue may have multiplicity > 1  Several possible solutions  Happens, for example, when graph is disconnected Problem #2: Rank accumulates at sinks.  Only sinks or nodes, from which a sink cannot be reached, can have nonzero rank mass. PageRank, Attempt #2

9. 9 Then:  r is a non-negative normalized left eigenvector of (B + 1e T ) with eigenvalue 1/ PageRank, Final Definition e = “rank source” vector  Standard setting: e(p) =  /n for all p (  < 1) 1 = the all 1’s vector

10. 10 Any nonzero eigenvalue of (B + 1e T ) may give a solution Pick r to be a principal left eigenvector of (B + 1e T ) Will show:  Principal eigenvalue has multiplicity 1, for any graph  There exists a non-negative left eigenvector Hence, PageRank always exists and is uniquely defined Due to rank source vector, rank no longer accumulates at sinks PageRank, Final Definition

11. 11 An Alternative View of PageRank: The Random Surfer Model When visiting a page p, a “random surfer”:  With probability 1 - d, selects a random outlink p  q and goes to visit q. (“focused browsing”)  With probability d, jumps to a random web page q. (“loss of interest”)  If p has no outlinks, assume it has a self loop. P: probability transition matrix:

12. 12 PageRank & Random Surfer Model Therefore, r is a principal left eigenvector of (B + 1e T ) if and only if it is a principal left eigenvector of P. Suppose: Then:

13. 13 PageRank & Markov Chains PageRank vector is normalized principal left eigenvector of (B + 1e T ). Hence, PageRank vector is also a principal left eigenvector of P Conclusion: PageRank is the unique stationary distribution of the random surfer Markov Chain. PageRank(p) = r(p) = probability of random surfer visiting page p at the limit. Note: “Random jump” guarantees Markov Chain is ergodic.

14. 14 HITS: Hubs and Authorities [Kleinberg, 1997] HITS: Hyperlink Induced Topic Search Main principle: every page p is associated with two scores:  Authority score: how “authoritative” a page is about the query’s topic Ex: query: “IR”; authorities: scientific IR papers Ex: query: “automobile manufacturers”; authorities: Mazda, Toyota, and GM web sites  Hub score: how good the page is as a “resource list” about the query’s topic Ex: query: “IR”; hubs: surveys and books about IR Ex: query: “automobile manufacturers”; hubs: KBB, car link lists

15. 15 Mutual Reinforcement HITS principles: p is a good authority, if it is linked by many good hubs. p is a good hub, if it points to many good authorities.

16. 16 HITS: Algebraic Form a: authority vector h: hub vector A: adjacency matrix Then: Therefore: a is principal eigenvector of A T A h is principal eigenvector of AA T

17. 17 Co-Citation and Bibilographic Coupling A T A: co-citation matrix  A T A p,q = # of pages that link both to p and to q.  Thus: authority scores propagate through co-citation. AA T : bibliographic coupling matrix  AA T p,q = # of pages that both p and q link to.  Thus: hub scores propagate through bibliographic coupling. p q p q

18. 18 Principal Eigenvector Computation E: n × n matrix | 1 | > | 2 | ≥ | 3 | … ≥ | n | : eigenvalues of E  Suppose 1 > 0 v 1,…,v n : corresponding eigenvectors Eigenvectors are form an orthornormal basis Input:  The matrix E  A unit vector u, which is not orthogonal to v 1 Goal: compute 1 and v 1

19. 19 The Power Method

20. 20 Why Does It Work? Theorem: As t  , w  c · v 1 (c is a constant) Convergence rate: Proportional to ( 2 / 1 ) t The larger the “spectral gap” 2 - 1, the faster the convergence.

21. 21 Spectral Methods in Information Retrieval

22. 22 Outline Motivation: synonymy and polysemy Latent Semantic Indexing (LSI) Singular Value Decomposition (SVD) LSI via SVD Why LSI works? HITS and SVD

23. 23 Synonymy and Polysemy Synonymy: multiple terms with (almost) the same meaning  Ex: cars, autos, vehicles  Harms recall Polysemy: a term with multiple meanings  Ex: java (programming language, coffee, island)  Harms precision

24. 24 Traditional Solutions Query expansion  Synonymy: OR on all synonyms Manual/automatic use of thesauri Too few synonyms: recall still low Too many synonyms: harms precision  Polysemy: AND on term and additional specializing terms Ex: +java +”programming language” Too broad terms: precision still low Too narrow terms: harms recall

25. 25 Syntactic Space D: document collection, |D| = n T: term space, |T| = m A t,d : “weight” of t in d (e.g., TFIDF) A T A: pairwise document similarities AA T : pairwise term similarities A m n terms documents

26. 26 Syntactic Indexing Index keys: terms Limitations  Synonymy (Near)-identical rows  Polysemy  Space inefficiency Matrix usually is not full rank Gap between syntax and semantics: Information need is semantic but index and query are syntactic.

27. 27 Semantic Space C: concept space, |C| = r B c,d : “weight” of c in d Change of basis Compare to wavelet and Fourier transforms B r n concepts documents

28. 28 Latent Semantic Indexing (LSI) [Deerwester et al. 1990] Index keys: concepts Documents & query: mixtures of concepts Given a query, finds the most similar documents Bridges the syntax-semantics gap Space-efficient  Concepts are orthogonal  Matrix is full rank Questions  What is the concept space?  What is the transformation from the syntax space to the semantic space?  How to filter “noise concepts”?

29. 29 Singular Values A: m×n real matrix Definition:  ≥ 0 is a singular value of A if there exist a pair of vectors u,v s.t. Av =  u and A T u =  v u and v are called singular vectors. Ex:  = ||A|| 2 = max ||x|| 2 = 1 ||Ax|| 2.  Corresponding singular vectors: x that maximizes ||Ax|| 2 and y = Ax / ||A|| 2. Note: A T Av =  2 v and AA T u =  2 u   2 is eigenvalue of A T A and AA T  u eigenvector of A T A  v eigenvector of AA T

30. 30 Singular Value Decomposition (SVD) Theorem: For every m×n real matrix A, there exists a singular value decomposition: A = U  V T   1 ≥ … ≥  r > 0 (r = rank(A)): singular values of A   = Diag(  1,…,  r )  U: column-orthonormal m×r matrix (U T U = I )  V: column-orthonormal n×r matrix (V T V = I ) A U VTVT ××  =

31. 31 Singular Values vs. Eigenvalues A = U  V T  1,…,  r : singular values of A   1 2,…,  r 2 : non-zero eigenvalues of A T A and AA T u 1,…,u r : columns of U  Orthonormal basis for span(columns of A)  Left singular vectors of A  Eigenvectors of A T A v 1,…,v r : columns of V  Orthonormal basis for span(rows of A)  Right singular vectors  Eigenvectors of AA T

32. 32 LSI as SVD A = U  V T  U T A =  V T u 1,…,u r : concept basis B =  V T : LSI matrix A d : d-th column of A B d : d-th column of B B d = U T A d B d [c] = u c T A d

33. 33 Noisy Concepts B = U T A =  V T B d [c] =  c v d [c] If  c is small, then B d [c] small for all d k = largest i s.t.  i is “large” For all c = k+1,…,r, and for all d, c is a low- weight concept in d Main idea: filter out all concepts c = k+1,…,r  Space efficient: # of index terms = k (vs. r or m)  Better retrieval: noisy concepts are filtered out across the board

34. 34 Low-rank SVD B = U T A =  V T U k = (u 1,…,u k ) V k = (v 1,…,v k )  k = upper-left k×k sub-matrix of  A k = U k  k V k T B k =  k V k T rank(A k ) = rank(B k ) = k

35. 35 Low Dimensional Embedding Forbenius norm: Fact: Therefore, if is small, then for “most” d,d’,. A k preserves pairwise similarities among documents  at least as good as A for retrieval.

36. 36 Computing SVD Compute singular values of A, by computing eigenvalues of A T A Compute U,V by computing eigenvectors of A T A and AA T Running time not too good: O(m 2 n + m n 2 )  Not practical for huge corpora Sub-linear time algorithms for estimating A k [Frieze,Kannan,Vempala 1998]

37. 37 HITS and SVD A: adjacency matrix of a web (sub-)graph G a: authority vector h: hub vector a is principal eigenvector of A T A h is principal eigenvector of AA T Therefore: a and h give A 1 : the rank-1 SVD of A Generalization: using A k, we can get k authority and hub vectors, corresponding to other topics in G.

38. 38 Why is LSI Better? [Papadimitriou et al. 1998] [Azar et al. 2001] LSI summary  Documents are embedded in low dimensional space (m  k)  Pairwise similarities are preserved  More space-efficient But why is retrieval better?  Synonymy  Polysemy

39. 39 Generative Model A corpus model M = (T,C,W,D)  T: Term space, |T| = m  C: Concept space, |C| = k Concept: distribution over terms  W: Topic space Topic: distribution over concepts  D: Document distribution Distribution over W × N A document d is generated as follows:  Sample a topic w and a length n according to D  Repeat n times: Sample a concept c from C according to w Sample a term t from T according to c

40. 40 Simplifying Assumptions Every document has a single topic (W = C) For every two concepts c,c’, ||c – c’|| ≥ 1 -  The probability of every term under a concept c is at most some constant .

41. 41 LSI Works A: m×n term-document matrix, representing n documents generated according to the model Theorem [Papadimitriou et al. 1998] With high probability, for every two documents d,d’,  If topic(d) = topic(d’), then  If topic(d)  topic(d’), then

42. 42 Proof For simplicity, assume  = 0 Want to show:  If topic(d) = topic(d’), A d k || A d’ k  If topic(d)  topic(d’), A d k  A d’ k D c : documents whose topic is the concept c T c : terms in supp(c)  Since ||c – c’|| = 1, T c ∩ T c’ = Ø A has non-zeroes only in blocks: B 1,…,B k, where B c : sub-matrix of A with rows in T c and columns in D c A T A is a block diagonal matrix with blocks B T 1 B 1,…, B T k B k (i,j)-th entry of B T c B c : term similarity between i-th and j-th documents whose topic is the concept c B T c B c : adjacency matrix of a bipartite (multi-)graph G c on D c

43. 43 Proof (cont.) G c is a “random” graph First and second eigenvalues of B T c B c are well separated For all c,c’, second eigenvalue of B T c B c is smaller than first eigenvalue of B T c’ B c’ Top k eigenvalues of A T A are the principal eigenvalues of B T c B c for c = 1,…,k Let u 1,…,u k be corresponding eigenvectors For every document d on topic c, A d is orthogonal to all u 1,…,u k, except for u c. A k d is a scalar multiple of u c.

44. 44 Extensions [Azar et al. 2001] A more general generative model Explain also improved treatment of polysemy

45. 45 End of Lecture 5