Informatics and Mathematical Modelling / Intelligent Signal Processing
Non-negative Tensor Decompositions
Morten Mørup
Informatics and Mathematical Modeling
Intelligent Signal Processing
Technical University of Denmark
Morten Mørup
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Informatics and Mathematical Modelling / Intelligent Signal Processing
Sæby, May 22-2006
Parts of the work done in
collaboration with
Lars Kai Hansen, Professor
Sidse M. Arnfred, Dr. Med. PhD
Mikkel N. Schmidt, Stud. PhD
Department of Signal Processing
Informatics and Mathematical Modeling,
Technical University of Denmark
Cognitive Research Unit
Hvidovre Hospital
University Hospital of Copenhagen
Department of Signal Processing
Informatics and Mathematical Modeling,
Technical University of Denmark
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Overview
 Non-negativity Matrix Factorization
(NMF)
 Sparse coding NMF
(SNMF)
 Sparse Higher Order Non-negative Matrix
Factorization (HONMF)
 Sparse Non-negative Tensor double deconvolution
(SNTF2D)
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Factor Analysis
tests
Wd
d
Int.

tests
Subjects
Subjects
Int.
Spearman ~1900
Hd
VWH
Vtests x subjects  Wtests x intelligencesHintelligencesxsubject
Non-negative Matrix Factorization (NMF):
VWH s.t. Wi,d,Hd,j0
(~1970 Lawson, ~1995 Paatero, ~2000 Lee & Seung)
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The idea behind multiplicative updates
Positive term
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Negative term
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Non-negative matrix factorization (NMF)
(Lee & Seung - 2001)
NMF gives Part based representation
(Lee & Seung – Nature 1999)
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The NMF decomposition is not unique
Simplical Cone
~~
V  WH  (WP)(P -1 H)  WH
Positive Orthant
Convex Hull
z
z
z
y
y
y
x
x
x
NMF only unique when data adequately spans the positive orthant
(Donoho & Stodden - 2004)
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Sparse Coding NMF (SNMF)
(Mørup & Schmidt, 2006)
(Eggert & Körner, 2004)
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Swimmer Articulations
Illustration (the swimmer problem)
V ( Articulation pixel )  W ( ArticulationExpression) H ( Expression pixel )
True Expressions
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NMF Expressions
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SNMF Expressions
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Why sparseness?
 Ensures uniqueness
 Eases interpretability
(sparse representation  factor effects pertain to fewer dimensions)
 Can work as model selection
(Sparseness can turn off excess factors by letting them become zero)
 Resolves over complete representations
(when model has many more free variables than data points)
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Extensions to tensors
Factor Analysis
TUCKER
TUCKER
PARAFAC
A 1
A d1
Wd
d
Hd
D
Vi1i2   Wi1d H i2 d
d 1
d
A3
A d3
A d2 
D
Vi1i2i3   A i11d A i22d A i33d
d 1
=
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Vi1i2i3 
11
G
J3
J2
J1
j3
j2
j1
A2 
Vi1i2i3  G j1 j2 j3 A i11j1 A i22j2 A i33j3
J1
G
j1  j2  j3
j1 j1 j1
A i11j1 A i22j1 A i33j1
Informatics and Mathematical Modelling / Intelligent Signal Processing
Uniqueness
 Although PARAFAC in general is unique under mild conditions, the
proof of uniqueness by Kruskal is based on k-rank*. However, the krank does not apply for non-negativity**.
 TUCKER model is not unique, thus no guaranty of uniqueness.
Imposing sparseness useful in order to achieve unique decompositions
Tensor decompositions known to have problems with degeneracy,
however when imposing non-negativity degenerate solutions can’t occur***
*) k-rank: The maximum number of columns chosen by random of a matrix certain to be linearly independent.
**) L.-H. Lim and G.H. Golub, 2006.
***) See L.-H. Lim - http://www.etis.ensea.fr/~wtda/Articles/wtda-nnparafac-slides.pdf
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Example why Non-negative PARAFAC isn’t unique
1  1 ( 2 ) 1 0 ( 3) 1 0
A (1)  
, A  1 1 , A  2 1
1 1 




1  1
1 1 0 1 1
X1  
diag
(

0 ) 1 1  1 1
1 1 
  
 

T
1  1
2 1 0 2 1
X2  
diag (   ) 

 

1 1 
1  1 1 2 3
Kruskal condition : K A  K B  K C  2 F  2 satisfied
T
Non  negative rank  3 :
1 1
1 1 1 0 0 0
I : X1  
,
X

2

1 1  1 0  0 2
1 1

 
 

1 0 0 1 0 0
1 0 0 1 0 0
II : X1  


, X 2  2




  3

1 0 0 0 0 1
1 0 0 0 0 1
1 ½ 
0 1 
0 0 
1 ½  0 0
III : X1  

½

½
,
X

2
2

0 0 
0 1 
1 ½   2 0 1
1 ½ 





 

1 ½ 
0 1
1 ½  0 0
IV : X1  
 ½
, X 2  2


  2

1 ½ 
0 1
1 ½  0 1
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PARAFAC model estimation

A  B  A1  B1 A2  B2  A J  B J

V1  A 1Z 1

Z 1  A 3  A 2 
V
V3   A 3Z 3
A d1
d

Z 3  A 2   A 1
A d3


T
T
A d2 
V2   A 2 Z 2 

D
Vi1i2i3   A i11d A i22d A i33d
Z 2   A 3  A 1
d 1

T
Thus, the PARAFAC model is by the matricizing operation
estimated straight forward from regular NMF estimation by interchanging
W with A and H with Z.
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TUCKER model estimation
V1  A 1Z 1
TUCKER

Z 1  G (1) A 3  A 2 
A d1
V2   A 2 Z2 
A
1

Z3  G 3 A 2   A 1

Z
A3
J2
J1
j3
j2
j1
T
T
vecV   vecG (A3  A2   A1 )
Vi1i2i3  G j1 j2 j3 A i11j1 A i22j2 A i33j3
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 G 2  A  A

A2 
J3
3 
V3   A 3Z3
G
T
1
2 


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Algorithms for Non-negative TUCKER
(PARAFAC follows by setting C=I)
(Mørup et al. 2006)
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Application of Non-negative TUCKER and
PARAFAC
Non-negative TUCKER in the following called
HONMF
(Higher order non-negative matrix factorization)
Non-negative PARAFAC called NTF
(Non-negative tensor factorization)
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Continuous Wavelet transform
Absolute value of wavelet coefficient
frequency
Complex Morlet wavelet
- Real part - Complex part
time
time
 e i
Captures frequency changes through time
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channel
Channel x Time-Frequency x Subjects
time-frequency
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Results
HONMF with sparseness, above imposed on the core can
be used for model selection -here indicating the PARAFAC
model is the appropriate model to the data.
Furthermore, the HONMF gives a more part based hence easy
interpretable solution than the HOSVD.
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Evaluation of uniqueness
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Data of a Flow Injection Analysis (Nørrgaard, 1994)
HONMF with sparse core and mixing captures unsupervised
the true mixing and model order!
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Many of the data sets previously explored by the Tucker model are nonnegative and could with good reason be decomposed under constraints of
non-negativity on all modalities including the core.
BatchSpectreTime
X Strength
(Smilde et al. 1999,2004, Andersson & Bro 1998, Nørgard & Ridder 1994)
 Spectroscopy data
 Web mining
UsersQueriesWeb pages
X Click
counts
 Image Analysis
PeopleViewsIlluminationsExpressionsPixels
X Image
Intensity
 Semantic Differential Data
JudgesMusic PiecesScales
X Grade
(Sun et al., 2004)
(Vasilescu and Terzopoulos, 2002, Wang and Ahuja, 2003, Jian and Gong, 2005)
(Murakami and Kroonenberg, 2003)
 And many more……
Hopefully, the devised algorithms for sparse non-negative TUCKER will
prove useful
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Conclusion
 HONMF and NTF not in general unique, however
when imposing sparseness uniqueness can be
achieved.
 Algorithms devised for LS and KL able to impose
sparseness on any combination of modalities
 The HONMF decompositions more part based hence
easier to interpret than other Tucker decompositions
such as the HOSVD.
 Imposing sparseness can work as model selection
turning of excess components
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Released 14th September 2006
ERPWAVELAB
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Sparse Non-negative Tensor Factor
double deconvolution for music
separation and transcription
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The ‘ideal’ Log-frequency Magnitude Spectrogram
of an instrument
 Different notes played by an
instrument corresponds on a
logarithmic frequency scale to a
translation of the same harmonic
structure of a fixed temporal pattern
Tchaikovsky: Violin Concert in D Major
3200
1600
800
Frequency [Hz]
Mozart Sonate no,. 16 in C Major
400
200
0
0.5
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1.5
2
Time [s]
2.5
3
3.5
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NMF 2D deconvolution (NMF2D1): The Basic Idea
 Model a log-spectrogram of polyphonic music by an
extended type of non-negative matrix factorization:
– The frequency signature of a specific note played by an
instrument has a fixed temporal pattern (echo)
 model convolutive in time
– Different notes of same instrument has same time-logfrequency signature but varying in fundamental frequency
(shift)
 model convolutive in the log-frequency axis.
(1Mørup & Scmidt, 2006)
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Vi , j  Λ 
V
Wi  ,d , H
H d , j  ,
W
8
4
0
 , , d

Understanding the NMF2D Model
1600
800
400
200
0246

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0.2
29
0.6
0.4
Time [s]
0.8
Frequency [Hz]
3200
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The NMF2D has inherent ambiguity between the
structure in W and H
To resolve this ambiguity sparsity is imposed
on H to force ambiguous structure onto W
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Real music example of how imposing sparseness
resolves the ambiguity between W and H
NMF2D
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SNMF2D
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Mozart Sonate no. 16 in C Major


Tchaikovsky: Violin Concert in D Major

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
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Sparse Non-negative Tensor Factor 2D deconvolution (SNTF2D)
(Extension of Fitzgerald et al. 2005, 2006 to form a sparse double deconvolution)
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Stereo recording of ”Fog is Lifting” by Carl Nielsen
Stereo Channel 2
Stereo Channel 1
Log-Spectrogram Channel 1
Log-Spectrogram Channel 2
22 kHz
50 Hz
50 Hz
0.9071
25.9 ms
0.420
6850
Estimated Harp
22 kHz
22 kHz
50 Hz
50 Hz
25.9 ms
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25.9 ms
0.7286
22 kHz
Estimated Flute
25.9 ms
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Applications
 Applications
–
–
–
–
Source separation.
Music information retrieval.
Automatic music transcription (MIDI compression).
Source localization (beam forming)
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References
Carroll, J. D. and Chang, J. J. Analysis of individual differences in multidimensional scaling via an N-way generalization of "Eckart-Young" decomposition, Psychometrika 35 1970 283—319
Donoho, D. and Stodden, V. When does non-negative matrix factorization give a correct decomposition into parts? NIPS2003
Eggert, J. and Korner, E. Sparse coding and NMF. In Neural Networks volume 4, pages 2529-2533, 2004
Eggert, J et al Transformation-invariant representation and nmf. In Neural Networks, volume 4 , pages 535-2539, 2004
Fiitzgerald, D. et al. Non-negative tensor factorization for sound source separation. In proceedings of Irish Signals and Systems Conference, 2005
FitzGerald, D. and Coyle, E. C Sound source separation using shifted non.-negative tensor factorization. In ICASSP2006, 2006
Fitzgerald, D et al. Shifted non-negative matrix factorization for sound source separation. In Proceedings of the IEEE conference on Statistics in Signal Processing. 2005
Kruskal, J.B. Three-way analysis: rank and uniqueness of trilinear decompostions, with application to arithmetic complexity and statistics. Linear Algebra Appl., 18: 95-138, 1977
Harshman, R. A. Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis},UCLA Working Papers in Phonetics 16 1970 1—84
Harshman, Richard A.Harshman and Hong, Sungjin Lundy, Margaret E. Shifted factor analysis—Part I: Models and properties J. Chemometrics (17) pages 379–388, 2003
Lathauwer, Lieven De and Moor, Bart De and Vandewalle, Joos MULTILINEAR SINGULAR VALUE DECOMPOSITION.SIAM J. MATRIX ANAL. APPL.2000 (21)1253–1278
Lee, D.D. and Seung, H.S. Algorithms for non-negative matrix factorization. In NIPS, pages 556-462, 2000
Lee, D.D and Seung, H.S. Learning the parts of objects by non-negative matrix factorization, NATURE 1999
Lim, Lek-Heng - http://www.etis.ensea.fr/~wtda/Articles/wtda-nnparafac-slides.pdf
Lim, L.-H. and Golub, G.H., "Nonnegative decomposition and approximation of nonnegative matrices and tensors," SCCM Technical Report, 06-01, forthcoming, 2006.
Murakami, Takashi and Kroonenberg, Pieter M. Three-Mode Models and Individual Differences in Semantic Differential Data, Multivariate Behavioral Research(38) no. 2 pages 247-283, 2003
Mørup, M. and Hansen, L.K.and Arnfred, S.M.Decomposing the time-frequency representation of EEG using nonnegative matrix and multi-way factorization Technical report, Institute for Mathematical
Modeling, Technical University of Denmark, 2006b
Mørup, M., Hansen, L. K., Arnfred, S. M., ERPWAVELAB A toolbox for multi-channel analysis of time-frequency transformed event related potentials, Journal of Neuroscience Methods, vol. 161, pp. 361-368,
2007a
Mørup, M., Hansen, L. K., Parnes, Josef, Hermann, C, Arnfred, S. M., Parallel Factor Analysis as an exploratory tool for wavelet transformed event-related EEG Neuroimage NeuroImage 29 938 – 947, 2006a
Mørup, M., Schmidt, M. N., Hansen, L. K., Shift Invariant Sparse Coding of Image and Music Data, submitted, JMLR, 2007b
Mørup, M., Hansen, L. K., Arnfred, S. M., Algorithms for Sparse Non-negative TUCKER, Submitted Neural Computation, 2006e
Mørup, M. and Hansen, L.K.and Arnfred, S.M.Decomposing the time-frequency representation of EEG using nonnegative matrix and multi-way factorization Technical report, Institute for Mathematical
Modeling, Technical University of Denmark, 2006a
Schmidt, M.N. and Mørup, M. Non-negative matrix factor 2D deconvolution for blind single channel source separation. In ICA2006, pages 700-707, 2006d
Nørgaard, L and Ridder, C.Rank annihilation factor analysis applied to flow injection analysis with photodiode-array detection Chemometrics and Intelligent Laboratory Systems 1994 (23) 107-114
Schmidt, M.N. and Mørup, M. Sparse Non-negative Matrix Factor 2-D Deconvolution for Automatic Transcription of Polyphonic Music, Technical report, Institute for Mathematical Modelling, Tehcnical
University of Denmark, 2005
Smaragdis, P. Non-negative Matrix Factor deconvolution; Extraction of multiple sound sources from monophonic inputs. International Symposium on independent Component Analysis and Blind Source
Separation (ICA)W
Smilde, Age K. Smilde and Tauller, Roma and Saurina, Javier and Bro, Rasmus, Calibration methods for complex second-order data Analytica Chimica Acta 1999 237-251
Sun, Jian-Tao and Zeng, Hua-Jun and Liu, Huanand Lu Yuchang and Chen Zheng CubeSVD: a novel approach to personalized Web search WWW '05: Proceedings of the 14th international conference on World
Wide Web pages 382—390, 2005
Tamara G. Kolda Multilinear operators for higher-order decompositions technical report Sandia national laboratory 2006 SAND2006-2081.
Tucker, L. R. Some mathematical notes on three-mode factor analysis Psychometrika 31 1966 279—311
Welling, M. and Weber, M. Positive tensor factorization. Pattern Recogn. Lett. 2001
Vasilescu , M. A. O. and Terzopoulos , Demetri Multilinear Analysis of Image Ensembles: TensorFaces, ECCV '02: Proceedings of the 7th European Conference on Computer Vision-Part I, 2002
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