Clustering and Factorization using Apache SystemML by Prithviraj Sen

7
Matrix Factorization Algorithms in Apache SystemML Prithviraj Sen

Transcript of Clustering and Factorization using Apache SystemML by Prithviraj Sen

Page 1: Clustering and Factorization using Apache SystemML by Prithviraj Sen

MatrixFactorizationAlgorithmsinApacheSystemML

PrithvirajSen

Page 2: Clustering and Factorization using Apache SystemML by Prithviraj Sen

ApplicationsofMatrixFactorization

• NetflixPrize• Givenratingsdatapredictwhatmoviesuserswillwatch

1 3 4 ? ? ?

? 3 5 ? ? 5

? ? 4 5 ? 5

? ? 3 ? ? ?

? ? 3 ? ? ?

2 ? ? 2 ? 2

? ? ? ? 5 ?

? 2 1 ? ? 1

? 3 ? ? 3 ?

1 ? ? ? ? ?

17,700 movies

480,000users

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Applications:PartsBasedDecomposition

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LeastSquaresMatrixFactorization

• ApproximateVusingLRmin∑ (𝑣$%−𝑙$′𝑟%)2($,%)

• Leadstotheverywellknownalternatingleastsquaresalgorithm• Onlyrequiressolvingleastsquares• Embarassingly parallel

V L

Rf

f

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ALSinDML

parfor

parfor

Directsolvingleastsquares

Directsolvingleastsquares

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PoissonMatrixFactorization(NMF)

• Suitableifyouarelookingfornon-negativefactors𝑣 = 𝑒/01𝑙𝑟𝑛/𝑛!

• LeadstothewellknownGeneralizedKL-Divergencemax∑ (𝑛𝑖𝑗log𝑙𝑖′𝑟𝑗 − 𝑙𝑖′𝑟𝑗)$%

• Wellknownupdateequationsexist*

*“Generalized NonnegativeMatrixApproximationswithBregman Divergences” byDhillon andSra inNIPS2005.* “DistributedNonnegativeMatrixFactorizationforWeb-ScaleDyadicDataAnalysisonMapReduce”byLiuetalinWWW2010.

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PNMFinDML

• Veryefficientupdatesusingonlylinearalgebra• UsesApacheSystemML’s wdivmm operator

wdivmm operator


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Factorize Factorization · 2 Factorize Factorization is however a more sophisticated property (sketched below) of which factorization is a basic instance. The content of factorization

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Shri Prithviraj Chavan Hon’ble Minister of State in Prime Minister Office Shri Prithviraj Chavan Hon’ble Minister of State in Prime Minister Office Power.

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