非负矩阵分解及在人脸识别的应用
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Non-Negative Matrix Factorization （NMF）
Paper:D.D.Lee andS.Seung,”Learning the parts of
objects by non-negative matrix factorization” Nature,vol.401,pp.788-791,1999
basis images
What don’t we like about PCA?
PCA involves adding up some basis images then subtracting others
Basis images aren’t physically intuitive Subtracting doesn’t make sense in context of
[2]D.D.Lee and H.S.Seung“Algorithms for non-negative Matrix factorization ”，in Proceedings of Neural Information Processing
Systems,2000. [3]S.Z.Li,X.Hou,H.J.Zhang,andQ.Cheng,“Learning spatially localized,parts-
Reportor: MaPeng
作者的相关信息
Daniel D. Lee, Ph.D. Associate Professor
Dept. of Electrical and Systems Engineering Dept. of Bioengineering (Secondary) GRASP (General Robotics, Automation, Sensing, Perception) Lab 203B Moore/6314 University of Pennsylvania 200 S. 33rd Street Philadelphia, PA 19104 215-898-8112 215-573-2068(FAX)
based representation”,Proc.IEEE Int.Conf.Computer Vision and Pattern Recognition,2001,pp.207-212
[4]J.Lu andY.-P.Tan,“Doubly weighted nonnegative matrix factorization for imbalanced face recognition”,Proc.IEபைடு நூலகம்E Int.Conf.Acoustics,Speech,andSignalProcessing,2009,pp.877¨C880
First 25 examples shown at right
Set consists of 19x19 centered face images
Faces
Basis Images:
– Rank: 49 – Iterations: 50
Faces
Original
x
=
Faces
Original
NMF is based on Gradient Descent
NMF:
VWH s.t. Wi,d,Hd,j0
Let C be a given cost function, then update the parameters according to:
The idea behind multiplicative
some applications
How do you subtract a face? What does subtraction mean in the context of document
classification? back
Non-negative Matrix Factorization
input image from the basis images 3. Dimension reduction
Mainly Discuss
PCA NMF LNMF FNMF WNMF
PCA V WH
Find a set of orthogonal basis images The reconstructed image is a linear combination of the
http://hebb.mit.edu/people/seung/
Problem Statement
Given a set of images: 1. Create a set of basis images that can be
linearly combined to create new images 2. Find the set of weights to reproduce every
NMF Basis Imagesnmf_basis
Only allowing adding of basis images makes intuitive sense
– Has physical analogue in neurons
Forcing the reconstruction coefficients to be positive leads to nice basis images
x
=
back
Example
4 1 3 C 6 2 4
A
1 2
3 4
1 1 0 B 1 0 1
C
AB
A*
1 1
1 0
0 1
Local non-negative matrix factorization
Letting U [uij ] BT B,V [vij ] HHT
MIT, 46-5065 43 Vassar St. Cambridge, MA 02139 voice: 617-252-1693 seung@mit.edu
Administrative assistant: Amy Dunn voice: 617-452-2694 fax: 617-452-2913 adunn@mit.edu
LNMF is aimed at learning local features by imposing the
following three additional constraints on the NMF basis:
back
LNMF_basis
Fisher non-negative matrix factorization
– To reconstruct images, all you can do is add in more basis images
– This leads to basis images that represent parts
Faces
Training set: 2429 examples
Email: ddlee@seas.upenn.edu http://www.seas.upenn.edu/~ddlee/
H. Sebastian Seung
Professor of Computational Neuroscience, MIT Investigator, Howard Hughes Medical Institute
updates
Positive term Negative term
The NMF decomposition is not unique
V WH (WP)(P -1H) W~ H~
~~
V WH (WP)(P1H ) W H
NMF only unique when data adequately spans the positive orthant (Donoho & Stodden - 2004)
Like PCA, except the coefficients in the linear combination cannot be negative
Non-negative matrix factorization (NMF)
(Lee & Seung - 2001)
NMF gives Part based representation (Lee & Seung – Nature 1999)
V WH
back
Weighted NMF
back
结论及未来工作
综上所述，非负矩阵分解是一种的提取 图像局部特征信息的有效的方法，目前 在很多领域得到广泛应用，值得我们关 注。
问题 （1）非平衡样本集识别率低的问题 （2）权重选取问题
参考文献
[1]D.D.Lee and H.S.Seung,“Learning the parts of objects by non-negative matrix factorization”, Nature,vol.401,pp.788-791,1999
Paper:D.D.Lee andS.Seung,”Learning the parts of
objects by non-negative matrix factorization” Nature,vol.401,pp.788-791,1999
basis images
What don’t we like about PCA?
PCA involves adding up some basis images then subtracting others
Basis images aren’t physically intuitive Subtracting doesn’t make sense in context of
[2]D.D.Lee and H.S.Seung“Algorithms for non-negative Matrix factorization ”，in Proceedings of Neural Information Processing
Systems,2000. [3]S.Z.Li,X.Hou,H.J.Zhang,andQ.Cheng,“Learning spatially localized,parts-
Reportor: MaPeng
作者的相关信息
Daniel D. Lee, Ph.D. Associate Professor
Dept. of Electrical and Systems Engineering Dept. of Bioengineering (Secondary) GRASP (General Robotics, Automation, Sensing, Perception) Lab 203B Moore/6314 University of Pennsylvania 200 S. 33rd Street Philadelphia, PA 19104 215-898-8112 215-573-2068(FAX)
based representation”,Proc.IEEE Int.Conf.Computer Vision and Pattern Recognition,2001,pp.207-212
[4]J.Lu andY.-P.Tan,“Doubly weighted nonnegative matrix factorization for imbalanced face recognition”,Proc.IEபைடு நூலகம்E Int.Conf.Acoustics,Speech,andSignalProcessing,2009,pp.877¨C880
First 25 examples shown at right
Set consists of 19x19 centered face images
Faces
Basis Images:
– Rank: 49 – Iterations: 50
Faces
Original
x
=
Faces
Original
NMF is based on Gradient Descent
NMF:
VWH s.t. Wi,d,Hd,j0
Let C be a given cost function, then update the parameters according to:
The idea behind multiplicative
some applications
How do you subtract a face? What does subtraction mean in the context of document
classification? back
Non-negative Matrix Factorization
input image from the basis images 3. Dimension reduction
Mainly Discuss
PCA NMF LNMF FNMF WNMF
PCA V WH
Find a set of orthogonal basis images The reconstructed image is a linear combination of the
http://hebb.mit.edu/people/seung/
Problem Statement
Given a set of images: 1. Create a set of basis images that can be
linearly combined to create new images 2. Find the set of weights to reproduce every
NMF Basis Imagesnmf_basis
Only allowing adding of basis images makes intuitive sense
– Has physical analogue in neurons
Forcing the reconstruction coefficients to be positive leads to nice basis images
x
=
back
Example
4 1 3 C 6 2 4
A
1 2
3 4
1 1 0 B 1 0 1
C
AB
A*
1 1
1 0
0 1
Local non-negative matrix factorization
Letting U [uij ] BT B,V [vij ] HHT
MIT, 46-5065 43 Vassar St. Cambridge, MA 02139 voice: 617-252-1693 seung@mit.edu
Administrative assistant: Amy Dunn voice: 617-452-2694 fax: 617-452-2913 adunn@mit.edu
LNMF is aimed at learning local features by imposing the
following three additional constraints on the NMF basis:
back
LNMF_basis
Fisher non-negative matrix factorization
– To reconstruct images, all you can do is add in more basis images
– This leads to basis images that represent parts
Faces
Training set: 2429 examples
Email: ddlee@seas.upenn.edu http://www.seas.upenn.edu/~ddlee/
H. Sebastian Seung
Professor of Computational Neuroscience, MIT Investigator, Howard Hughes Medical Institute
updates
Positive term Negative term
The NMF decomposition is not unique
V WH (WP)(P -1H) W~ H~
~~
V WH (WP)(P1H ) W H
NMF only unique when data adequately spans the positive orthant (Donoho & Stodden - 2004)
Like PCA, except the coefficients in the linear combination cannot be negative
Non-negative matrix factorization (NMF)
(Lee & Seung - 2001)
NMF gives Part based representation (Lee & Seung – Nature 1999)
V WH
back
Weighted NMF
back
结论及未来工作
综上所述，非负矩阵分解是一种的提取 图像局部特征信息的有效的方法，目前 在很多领域得到广泛应用，值得我们关 注。
问题 （1）非平衡样本集识别率低的问题 （2）权重选取问题
参考文献
[1]D.D.Lee and H.S.Seung,“Learning the parts of objects by non-negative matrix factorization”, Nature,vol.401,pp.788-791,1999