School of Computer Science Carnegie Mellon University Dept. of ECE University of Minnesota ParCube:...

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School of Computer ScienceCarnegie Mellon University

Dept. of ECEUniversity of Minnesota

ParCube: Sparse Parallelizable Tensor Decompositions

Evangelos E. Papalexakis1, Christos Faloutsos1, Nikos Sidiropoulos2

1Carnegie Mellon University, School of Computer Science2University of Minnesota, ECE Department

European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML

PKDD), Bristol, UK, September 24th-28th, 2012.

Evangelos Papalexakis (CMU) – ECML-PKDD 2012

Outline

• Introduction Problem Statement Method Experiments Conclusions

Introduction• Facebook has ~800 Million users

Evolves over time How do we spot interesting patterns & anomalies in this very large

network?

Introduction

• Suppose we have Knowledge Base data E.g. Read the Web Project at CMU

Subject – verb – object triplets, mined from the web Many gigabytes or terabytes of data! How do we find potential new synonyms to a

word using this knowledge base?

Introduction to Tensors

• Tensors are multidimensional generalizations of matrices Previous problems can be formulated as tensors! Time-evolving graphs/social networks, Multi-aspect

data (e.g. subject, object, verb)• Focus on 3-way tensors

Can be viewed as Data cubes

Indexed by 3variables (IxJxK)

subject

object

Introduction to Tensors

• PARAFAC decomposition Decompose a tensor into sum of outer products/rank 1

tensors Each rank 1 tensor is a different group/”concept” “Similar” to the Singular Value Decomposition in the

matrix case

Store the factor vectors ai, bi, ci as columns of matrices A, B, C

subject

object“leaders/CEOs” “products”

Outline

Introduction• Problem Statement Method Experiments Conclusions

Why not PARAFAC?

• Today’s datasets are in the orders of terabytes e.g. Facebook has ~ 800 Million users!

• Explosive complexity/run time for truly large datasets!

• Also, data is very sparse We need the decomposition factors to be sparse

Better interpretability / less noise Can do multi-way soft co-clustering this way!

PARAFAC is dense!

Problem Statement

• Wish-list: Significantly drop the dimensionality

Ideally 1 or more orders of magnitude Parallelize the computation

Ideally split the problem into independent parts and run in parallel

Yield sparse factors Don’t loose much in the process

Previous work

• A.H. Phan et al. Block decomposition for very large-scale nonnegative tensor factorization

Partition & merge parallel algorithm for NN PARAFAC No sparsity

• Q. Zhang et al. A parallel nonnegative tensor factorization algorithm for mining global climate data.

• D. Nion et al. Adaptive algorithms to track the parafac decomposition of a third-order tensor & J. Sun et al. Beyond streams and graphs: dynamic tensor analysis

Tensor is a stream, both methods seek to track the decomposition• C.E. Tsourakakis Mach: Fast randomized tensor decompositions & J. Sun et al.

Multivis:Content- based social network exploration through multi-way visual analysis Sampling based TUCKER models.

• E.E. Papalexakis et al. Co-clustering as multilinear decomposition with sparse latent factors.

Sparse PARAFAC algorithm applied to co-clustering

Evangelos Papalexakis (CMU) – ECML-PKDD 2012 10

None combines all

requirements!

Our proposal

• We introduce PARCUBE and set the following goals:• Goal 1: Fast

Scalable & parallelizable• Goal 2: Sparse

Ability to yield sparse latent factors and a sparse tensor approximation

• Goal 3: Accurate provable correctness in merging partial results, under

appropriate conditions

Outline

Introduction Problem Statement• Method Experiments Conclusions

PARCUBE: The big picture

• Sampling selects small portion of indices• PARAFAC vectors ai bi ci will be sparse by construction

Break up tensor into small piecesusing sampling

Fit dense PARAFAC decomposition on small sampled tensors

Match columns and distribute non-zero values to appropriate indices in original (non-sampled) space

The PARCUBE method

• Key ideas: Use biased sampling to sample rows, cols & fibers Sampling weight During sampling, always keep a common portion of

indices across samples For each smaller tensor, do the PARAFAC

decomposition. Need to specify 2 parameters:

Sampling rate: s Initial dimensions I, J, K I/s, J/s, K/s

Number of repetitions / different sampled tensors: r

Putting the pieces togetherDetails

• Say we have matrices As from each sample• Possibly have re-ordering of factors• Each matrix corresponds to different sampled index set of the

original index space• All factors share the “upper” part (by construction)

Proposition: Under mild conditions, the algorithm will stitch components correctly & output what exact PARAFAC would

Proof on paper

Outline

Introduction Problem Statement Method• Experiments Conclusions

Experiments

• We use the Tensor Toolbox for Matlab PARAFAC for baseline and core implementation

• Evaluation of performanceAlgorithm correctnessExecution speedupFactor sparsity

Experiments – Correctness for multiple repetitions

• Relative cost = PARCUBE approximation cost / PARAFAC approximation cost

• The more samples we get, the closer we are to exact PARAFAC• Experimental validation of our theoretical result.

Experiments - Correctness & Speedup for 1 repetition

• Relative cost = PARCUBE approximation cost / PARAFAC approximation cost• Speedup = PARAFAC execution time / PARCUBE execution time • Extrapolation to parallel execution for 4 repetitions yields 14.2x speedup

(and improves accuracy)

Experiments – Correctness & Sparsity

• Output size = NNZ(A) + NNZ(B) + NNZ(C)• 90% sparser than PARAFAC while maintaining the

same approximation error

Same as PARAFAC

Experiments

• Knowledge Discovery ENRON email/social network 186×186×44 Network traffic data (LBNL) 65170 × 65170 ×

65327 FACEBOOK Wall posts 63891 × 63890 × 1847 Knowledge Base data (Never Ending Language

Learner – NELL) 14545 × 14545 × 28818

Discovery - ENRON

• Who-emailed-whom data from the ENRON email dataset. Spans 44 months 184×184×44 tensor We picked s = 2, r = 4

• We were able to identify social cliques and spot spikes that correspond to actual important events in the company’s timeline

Discovery – LBNL Network Data

• Network traffic data of form (src IP, dst IP, port #) 65170 × 65170 × 65327 tensor We pick s = 5, r = 10

• We were able to identify a possible Port Scanning Attack

Discovery – FACEBOOK Wall posts

• Small portion of Facebook’s users 63890 users for 1847 days Picked s = 100, r = 10

• Data in the form (Wall owner, poster, timestamp)• Downloaded from http://socialnetworks.mpi-sws.org/data-wosn2009.html• We were able to identify a birthday-like event.

1 Wall

Discovery - NELL

• Knowledge base data• Taken from the Read The Web project at CMU

http://rtw.ml.cmu.edu/rtw/ Special thanks to Tom Mitchell for the data.• Noun phrase x Context x Noun phrase triplets

e.g. ‘Obama’ – ‘is’ – ‘the president of the United States’

• Discover words that may be used in the same context• We picked s = 500, r = 10.

Outline

Introduction Problem Statement Method Experiments• Conclusions

Conclusions

Goal 1: Fast Scalable & parallelizable

Goal 2: Sparse Ability to yield sparse latent factors and a sparse tensor

approximation Goal 3: Accurate

provable correctness in merging partial results, under appropriate conditions

Experiments that also demonstrate that

• Enables processing of tensors that don’t fit in memory• Interesting findings in diverse Knowledge Discovery settings

Evangelos Papalexakis (CMU) - ASONAM 2012 28

The End

Thank you!

Any questions?

Evangelos E. PapalexakisEmail: epapalex@cs.cmu.eduWeb: http://www.cs.cmu.edu/~epapalex

Christos FaloutsosEmail: christos@cs.cmu.eduWeb: http://www.cs.cmu.edu/~christos

Nicholas SidiropoulosEmail: nikos@umn.eduWeb: http://www.ece.umn.edu/users/nikos/

School of Computer Science Carnegie Mellon University Dept. of ECE University of Minnesota ParCube:...

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