Nonnegative Matrix Factorization via Rank-one Downdate

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Presentation transcript:

Nonnegative Matrix Factorization via Rank-one Downdate Ali Ghodsi Department of Statistics and Actuarial Science David R. Cheriton School of Computer Science University of Waterloo Joint work with Stephen Vavasis and Michael Biggs

Nonnegative Matrix Factorization

2 by 1965 560 by 1965 560 by 2 -2.19 -0.02 -3.19 1.02 20 by 28 20 by 28 2 by 1 2 by 1

Singular Value Decomposition (SVD)

History

History

History

History

History (Algorithms)

History (Algorithms)

First observation

Power method Computes the leading singular vectors/value (or eigenvector/value) of a matrix 1 2 while not converged 3 4 5 6 end

Naive approach to NMF using this observation 1 2 3 4 5 for all set 6 end for Without step 5, this will simply compute the SVD (Jordan's algorithm, Camille Jordan 1874. )

Rank-one Downdata (R1D)

Objective function

ApproxRankOneSubmatrix(A)

Modified power iteration: Demo Rank-1 submatrix A = Rank-1 submatrix

Modified power iteration: Demo v: 0.14 0.07 0.64 0.41 0.55 Rank-1 submatrix Rank-1 submatrix

Modified power iteration: Demo v: 0.0 0.0 0.64 0.41 0.55 Rank-1 submatrix Rank-1 submatrix

Modified power iteration: Demo v: 0.0 0.0 0.64 0.41 0.55 Rank-1 submatrix Rank-1 submatrix

Modified power iteration: Demo v: 0.0 0.0 0.64 0.41 0.55 0.16 0.21 0.22 0.44 0.74 0.20 Rank-1 submatrix u: Rank-1 submatrix

Modified power iteration: Demo v: 0.0 0.0 0.64 0.41 0.55 0.0 0.44 0.74 0.20 Rank-1 submatrix u: Rank-1 submatrix

Modified power iteration: Demo v: 0.0 0.0 0.64 0.41 0.55 0.0 0.44 0.74 0.20 Rank-1 submatrix u: Rank-1 submatrix

Modified power iteration: Demo v: 0.0 0.0 0.60 0.28 0.59 0.0 0.44 0.74 0.20 Rank-1 submatrix u: Rank-1 submatrix

Modified power iteration: Demo v: 0.0 0.0 0.60 0.28 0.59 0.0 0.44 0.74 0.20 Rank-1 submatrix u: Rank-1 submatrix

Modified power iteration: Demo v: 0.0 0.0 0.60 0.28 0.59 0.0 0.44 0.74 0.20 Rank-1 submatrix u: Rank-1 submatrix Zero-out!

Modified power iteration: Demo Rank-1 submatrix Anew =

Rank-one Downdata (R1D)

A simple model for text

Generating a corpus in the model

Theorem about text

LSI

R1D

Theorem about images

Experimental results

LSI

NMF-DIV

R1D

LSI

NMF_DIV

R1D

Thank you!