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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. Nonnegative Matrix Factorization via Rank-one Downdate.

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Nonnegative Matrix Factorization via Rank-one Downdate

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  1. 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 Nonnegative Matrix Factorization via Rank-one Downdate

  2. Nonnegative Matrix Factorization

  3. -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

  4. Singular Value Decomposition (SVD)

  5. History

  6. History

  7. History

  8. History

  9. History (Algorithms)

  10. History (Algorithms)

  11. First observation

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

  13. 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. )

  14. Rank-one Downdata (R1D)

  15. Objective function

  16. ApproxRankOneSubmatrix(A)

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

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

  19. Modified power iteration: Demo v: 0.0 0.00.64 0.41 0.55 Rank-1 submatrix Rank-1 submatrix

  20. Modified power iteration: Demo v: 0.0 0.00.64 0.41 0.55 Rank-1 submatrix Rank-1 submatrix

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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!

  27. Modified power iteration: Demo Rank-1 submatrix Anew =

  28. Rank-one Downdata (R1D)

  29. A simple model for text

  30. Generating a corpus in the model

  31. Theorem about text

  32. LSI

  33. R1D

  34. Theorem about images

  35. Experimental results

  36. LSI

  37. NMF-DIV

  38. R1D

  39. LSI

  40. NMF_DIV

  41. R1D

  42. Thank you!

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