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Nonnegative Matrix Factorization via Rank-one Downdate. Nonnegative Matrix Factorization. -2.19. -3.19. -0.02. 1.02. 2 by 1965. 560 by 1965. 560 by 2. 20 by 28. 2 by 1. 2 by 1. 20 by 28. Singular Value Decomposition (SVD). History. History. History. History. History (Algorithms).
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-2.19 -3.19 -0.02 1.02 2 by 1965 560 by 1965 560 by 2 20 by 28 2 by 1 2 by 1 20 by 28
Power method • Computes the leading singular vectors/value (or eigenvector/value) of a matrix 1 2 while not converged 3 4 5 6 end
Example • Computes the leading singular vectors/value (or eigenvector/value) of a matrix 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. )
Modified power iteration 1 2 while not converged 3 4 5 6 7 8 end
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 0.00.64 0.41 0.55 Rank-1 submatrix Rank-1 submatrix
Modified power iteration: Demo v: 0.0 0.00.64 0.41 0.55 Rank-1 submatrix Rank-1 submatrix
u: Modified power iteration: Demo v: 0.0 0.00.64 0.410.55 0.16 0.21 0.22 0.44 0.74 0.20 Rank-1 submatrix Rank-1 submatrix
u: Modified power iteration: Demo v: 0.0 0.0 0.64 0.41 0.55 0.0 0.0 0.0 0.44 0.74 0.20 Rank-1 submatrix Rank-1 submatrix
u: Modified power iteration: Demo v: 0.0 0.0 0.64 0.41 0.55 0.0 0.0 0.0 0.44 0.74 0.20 Rank-1 submatrix Rank-1 submatrix
u: Modified power iteration: Demo v: 0.0 0.0 0.60 0.28 0.59 0.0 0.0 0.0 0.44 0.74 0.20 Rank-1 submatrix Rank-1 submatrix
u: Modified power iteration: Demo v: 0.0 0.0 0.60 0.28 0.59 0.0 0.0 0.0 0.44 0.74 0.20 Rank-1 submatrix Rank-1 submatrix
u: Modified power iteration: Demo v: 0.0 0.0 0.60 0.28 0.59 0.0 0.0 0.0 0.44 0.74 0.20 Rank-1 submatrix Rank-1 submatrix Zero-out!
Modified power iteration: Demo Rank-1 submatrix Anew =
Rank-one Downdata (R1D) 1 2 3 4 5 end for
Information retrieval task TDT corpus (pilot study, v. 1.3, 1997): news articles Identified topics (first few columns of ): Experimental results Topic: president, clinton, house, white Topic: bosnian, serbs, bosnia, serb, nato, sarajevo, air, bihac Topic: haiti, military, aristide, haitian, troops, port, invasion, … Topic: simpson, defense, judge, case, jury, trial, angeles, los, court, … Topic: bill, today, senate, republicans, house, congress, republican, …
Experimental results • For topic: “OJ Simpson trial” (simpson, defense, judge, case, jury, trial)
