Title: Nonnegative Matrix Factorization via Rank-one Downdate
1Nonnegative 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
2Nonnegative Matrix Factorization
3(No Transcript)
42 by 1965
560 by 1965
560 by 2
20 by 28
2 by 1
2 by 1
20 by 28
5Singular Value Decomposition (SVD)
6History
7History
8History
9History
10History (Algorithms)
11History (Algorithms)
12First observation
13 Power method
- Computes the leading singular vectors/value (or
eigenvector/value) of a matrix
1 2 while not converged
3
4
5 6 end
14Naive 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. )
15Rank-one Downdata (R1D)
16Objective function
17ApproxRankOneSubmatrix(A)
18Modified power iteration Demo
Rank-1 submatrix
A
Rank-1 submatrix
19Modified power iteration Demo
v
0.14 0.07 0.64 0.41 0.55
Rank-1 submatrix
Rank-1 submatrix
20Modified power iteration Demo
v
0.0 0.0 0.64 0.41 0.55
Rank-1 submatrix
Rank-1 submatrix
21Modified power iteration Demo
v
0.0 0.0 0.64 0.41 0.55
Rank-1 submatrix
Rank-1 submatrix
22Modified 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
Rank-1 submatrix
23Modified 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
Rank-1 submatrix
24Modified 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
Rank-1 submatrix
25Modified 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
Rank-1 submatrix
26Modified 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
Rank-1 submatrix
27Modified 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
Rank-1 submatrix
Zero-out!
28Modified power iteration Demo
Rank-1 submatrix
Anew
29Rank-one Downdata (R1D)
30A simple model for text
31Generating a corpus in the model
32Theorem about text
33LSI
34R1D
35Theorem about images
36Experimental results
37LSI
38NMF-DIV
39R1D
40LSI
41NMF_DIV
42R1D
43