Nonnegative Matrix Factorization via Rankone Downdate

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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
• University of Waterloo

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Nonnegative Matrix Factorization
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(No Transcript)
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2 by 1965
560 by 1965
560 by 2

20 by 28
2 by 1
2 by 1
20 by 28
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Singular Value Decomposition (SVD)
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History
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History
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History
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History
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History (Algorithms)
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History (Algorithms)
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First observation
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Power method

• Computes the leading singular vectors/value (or
  eigenvector/value) of a matrix

1 2 while not converged
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4
5 6 end
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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. )

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Rank-one Downdata (R1D)
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Objective function
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ApproxRankOneSubmatrix(A)
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Modified power iteration Demo
Rank-1 submatrix
A
Rank-1 submatrix
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Modified power iteration Demo
v
0.14 0.07 0.64 0.41 0.55
Rank-1 submatrix
Rank-1 submatrix
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Modified power iteration Demo
v
0.0 0.0 0.64 0.41 0.55
Rank-1 submatrix
Rank-1 submatrix
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Modified power iteration Demo
v
0.0 0.0 0.64 0.41 0.55
Rank-1 submatrix
Rank-1 submatrix
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Modified power iteration Demo
v
0.0 0.0 0.64 0.41 0.55
Rank-1 submatrix
0.16 0.21 0.22 0.44 0.74 0.20

• u

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

• u

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

• u

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

• u

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

• u

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

• u

Rank-1 submatrix
Zero-out!
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Modified power iteration Demo
Rank-1 submatrix
Anew
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Rank-one Downdata (R1D)
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A simple model for text
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Generating a corpus in the model
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Theorem about text
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LSI
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R1D
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Theorem about images
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Experimental results
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LSI
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NMF-DIV
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R1D
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LSI
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NMF_DIV
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R1D
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• Thank you!