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CMU SCS : Multimedia Databases and Data Mining Lecture #21: Tensor decompositions C. Faloutsos

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CMU SCS 2 Must-read Material Tamara G. Kolda and Brett W. Bader. Tensor decompositions and applications. Technical Report SAND , Sandia National Laboratories, Albuquerque, NM and Livermore, CA, November 2007 Tensor decompositions and applications.

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CMU SCS 3 Outline Goal: ‘Find similar / interesting things’ Intro to DB Indexing - similarity search Data Mining

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CMU SCS 4 Indexing - Detailed outline primary key indexing secondary key / multi-key indexing spatial access methods fractals text Singular Value Decomposition (SVD) -… -Tensors multimedia...

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CMU SCS 5 Most of foils by Dr. Tamara Kolda (Sandia N.L.) csmr.ca.sandia.gov/~tgkolda Dr. Jimeng Sun (CMU -> IBM) 3h tutorial:

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CMU SCS 6 Outline Motivation - Definitions Tensor tools Case studies

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CMU SCS 7 Motivation 0: Why “matrix”? Why matrices are important?

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CMU SCS 8 Examples of Matrices: Graph - social network John PeterMaryNick... John Peter Mary Nick

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CMU SCS 9 Examples of Matrices: cloud of n-d points chol#blood# age..... John Peter Mary Nick...

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CMU SCS 10 Examples of Matrices: Market basket market basket as in Association Rules milkbreadchoc.wine... John Peter Mary Nick...

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CMU SCS 11 Examples of Matrices: Documents and terms Paper#1 Paper#2 Paper#3 Paper#4 dataminingclassif.tree...

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CMU SCS 12 Examples of Matrices: Authors and terms dataminingclassif.tree... John Peter Mary Nick...

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CMU SCS 13 Examples of Matrices: sensor-ids and time-ticks t1 t2 t3 t4 temp1temp2humid.pressure...

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CMU SCS 14 Motivation: Why tensors? Q: what is a tensor?

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CMU SCS 15 Motivation 2: Why tensor? A: N-D generalization of matrix: dataminingclassif.tree... John Peter Mary Nick... KDD’07

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CMU SCS 16 Motivation 2: Why tensor? A: N-D generalization of matrix: dataminingclassif.tree... John Peter Mary Nick... KDD’06 KDD’05 KDD’07

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CMU SCS 17 Tensors are useful for 3 or more modes Terminology: ‘mode’ (or ‘aspect’): dataminingclassif.tree... Mode (== aspect) #1 Mode#2 Mode#3

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CMU SCS 18 Motivating Applications Why matrices are important? Why tensors are useful? –P1: social networks –P2: web mining

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CMU SCS 19 P1: Social network analysis Traditionally, people focus on static networks and find community structures We plan to monitor the change of the community structure over time

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CMU SCS 20 P2: Web graph mining How to order the importance of web pages? –Kleinberg’s algorithm HITS –PageRank –Tensor extension on HITS (TOPHITS) context-sensitive hypergraph analysis

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CMU SCS 21 Outline Motivation – Definitions Tensor tools Case studies Tensor Basics Tucker PARAFAC

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CMU SCS Tensor Basics

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CMU SCS 23 Reminder: SVD –Best rank-k approximation in L2 A m n m n U VTVT

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CMU SCS 24 Reminder: SVD –Best rank-k approximation in L2 A m n + 1u1v11u1v1 2u2v22u2v2

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CMU SCS 25 Goal: extension to >=3 modes ~ I x R K x R A B J x R C R x R x R I x J x K +…+=

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CMU SCS 26 Main points: 2 major types of tensor decompositions: PARAFAC and Tucker both can be solved with ``alternating least squares’’ (ALS) Details follow

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CMU SCS Specially Structured Tensors

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CMU SCS 28 = U I x R V J x R W K x R R x R x R Specially Structured Tensors Tucker TensorKruskal Tensor I x J x K = U I x R V J x S W K x T R x S x T I x J x K Our Notation +…+= u1u1 uRuR v1v1 w1w1 vRvR wRwR “core”

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CMU SCS 29 Specially Structured Tensors Tucker TensorKruskal Tensor In matrix form: details

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CMU SCS Tensor Decompositions

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CMU SCS 31 Tucker Decomposition - intuition I x J x K ~ A I x R B J x S C K x T R x S x T author x keyword x conference A: author x author-group B: keyword x keyword-group C: conf. x conf-group : how groups relate to each other Needs elaboration!

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CMU SCS 32 Intuition behind core tensor 2-d case: co-clustering [Dhillon et al. Information-Theoretic Co- clustering, KDD’03]

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CMU SCS 33 m m n nl k k l eg, terms x documents

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CMU SCS 34 term x term-group doc x doc group term group x doc. group med. terms cs terms common terms med. doc cs doc

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CMU SCS 35

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CMU SCS 36 Tucker Decomposition Proposed by Tucker (1966) AKA: Three-mode factor analysis, three-mode PCA, orthogonal array decomposition A, B, and C generally assumed to be orthonormal (generally assume they have full column rank) is not diagonal Not unique Recall the equations for converting a tensor to a matrix I x J x K ~ A I x R B J x S C K x T R x S x T Given A, B, C, the optimal core is:

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CMU SCS 37 Outline Motivation – Definitions Tensor tools Case studies Tensor Basics Tucker PARAFAC

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CMU SCS 38 CANDECOMP/PARAFAC Decomposition CANDECOMP = Canonical Decomposition (Carroll & Chang, 1970) PARAFAC = Parallel Factors (Harshman, 1970) Core is diagonal (specified by the vector ) Columns of A, B, and C are not orthonormal If R is minimal, then R is called the rank of the tensor (Kruskal 1977) Can have rank( ) > min{I,J,K} ¼ I x R K x R A B J x R C R x R x R I x J x K +…+=

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CMU SCS 39 Tucker vs. PARAFAC Decompositions Tucker –Variable transformation in each mode –Core G may be dense –A, B, C generally orthonormal –Not unique PARAFAC –Sum of rank-1 components –No core, i.e., superdiagonal core –A, B, C may have linearly dependent columns –Generally unique I x J x K ~ A I x R B J x S C K x T R x S x T I x J x K +…+ ~ a1a1 aRaR b1b1 c1c1 bRbR cRcR IMPORTANT

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CMU SCS 40 Tensor tools - summary Two main tools –PARAFAC –Tucker Both find row-, column-, tube-groups –but in PARAFAC the three groups are identical To solve: Alternating Least Squares Toolbox: from Tamara Kolda:

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CMU SCS 41 Outline Motivation - Definitions Tensor tools Case studies

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CMU SCS 42 P1: Web graph mining How to order the importance of web pages? –Kleinberg’s algorithm HITS –PageRank –Tensor extension on HITS (TOPHITS)

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CMU SCS 43 Kleinberg’s Hubs and Authorities (the HITS method) Sparse adjacency matrix and its SVD: authority scores for 1 st topic hub scores for 1 st topic hub scores for 2 nd topic authority scores for 2 nd topic from to Kleinberg, JACM, 1999

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CMU SCS 44 authority scores for 1 st topic hub scores for 1 st topic hub scores for 2 nd topic authority scores for 2 nd topic from to HITS Authorities on Sample Data We started our crawl from and crawled 4700 pages, resulting in 560 cross-linked hosts.

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CMU SCS 45 Three-Dimensional View of the Web Observe that this tensor is very sparse! Kolda, Bader, Kenny, ICDM05

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CMU SCS 46 Topical HITS (TOPHITS) Main Idea: Extend the idea behind the HITS model to incorporate term (i.e., topical) information. authority scores for 1 st topic hub scores for 1 st topic hub scores for 2 nd topic authority scores for 2 nd topic from to term term scores for 1 st topic term scores for 2 nd topic

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CMU SCS 47 Topical HITS (TOPHITS) Main Idea: Extend the idea behind the HITS model to incorporate term (i.e., topical) information. authority scores for 1 st topic hub scores for 1 st topic hub scores for 2 nd topic authority scores for 2 nd topic from to term term scores for 1 st topic term scores for 2 nd topic

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CMU SCS 48 TOPHITS Terms & Authorities on Sample Data TOPHITS uses 3D analysis to find the dominant groupings of web pages and terms. authority scores for 1 st topic hub scores for 1 st topic hub scores for 2 nd topic authority scores for 2 nd topic from to term term scores for 1 st topic term scores for 2 nd topic Tensor PARAFAC w k = # unique links using term k

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CMU SCS GigaTensor: Scaling Tensor Analysis Up By 100 Times – Algorithms and Discoveries U Kang Christos Faloutsos School of Computer Science Carnegie Mellon University Evangelos Papalexakis Abhay Harpale

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CMU SCS P2: N.E.L.L. analysis NELL: Never Ending Language Learner –Q1: dominant concepts / topics? –Q2: synonyms for a given new phrase? 50 “Barrack Obama is the president of U.S.” “Eric Clapton plays guitar” (26M) (48M) NELL (Never Ending Language Learner) Nonzeros =144M (26M)

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CMU SCS A1: Concept Discovery Concept Discovery in Knowledge Base

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CMU SCS A1: Concept Discovery

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CMU SCS A2: Synonym Discovery Synonym Discovery in Knowledge Base a1a1 a2a2 aRaR … (Given) subject (Discovered) synonym 1 (Discovered) synonym 2

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CMU SCS A2: Synonym Discovery

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CMU SCS 55 Conclusions Real data are often in high dimensions with multiple aspects (modes) Matrices and tensors provide elegant theory and algorithms I x J x K +…+ ~ a1a1 aRaR b1b1 c1c1 bRbR cRcR

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CMU SCS 56 References Inderjit S. Dhillon, Subramanyam Mallela, Dharmendra S. Modha: Information-theoretic co-clustering. KDD 2003: T. G. Kolda, B. W. Bader and J. P. Kenny. Higher- Order Web Link Analysis Using Multilinear Algebra. In: ICDM 2005, Pages , November Jimeng Sun, Spiros Papadimitriou, Philip Yu. Window- based Tensor Analysis on High-dimensional and Multi- aspect Streams, Proc. of the Int. Conf. on Data Mining (ICDM), Hong Kong, China, Dec 2006

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