![Page 1: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/1.jpg)
Generalities about NMF Itakura-Saito NMF Variants
Itakura-Saito nonnegative matrix factorizationand friends for music signal decomposition
Cedric Fevotte
CNRS LTCI; Telecom ParisTech
ESI, ViennaDecember 2012
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 2: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/2.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Outline
Generalities about NMFConcept of NMFThe β-divergence: a unifying measure of fitMajorization-minimization algorithms
Itakura-Saito NMFStatistical modelPiano decomposition example
Variants of IS-NMFRegularized IS-NMFMultichannel IS-NMF
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 3: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/3.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Nonnegative matrix factorization (NMF)
Given a nonnegative matrix V of dimensions F × N, NMF is theproblem of finding a factorization
V ≈WH
where W and H are nonnegative matrices of dimensions F × Kand K × N, respectively.
K is usually chosen such that F K + K N << F N, hence reducingthe data dimension, but not always.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 4: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/4.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
An unsupervised part-based representation
Along VQ, PCA or ICA, NMF provides an unsupervised linearrepresentation of data
vn ≈ W hn
data vectors “explanatory variables” “regressors”“basis”, “dictionary” “expansion coefficients”
“patterns” “activation coefficients”
and W is learnt from the set of data vectors V = [v1 . . . vN ].
I nonneg. of W ensures interpretability of the dictionary(features wk and data vn belong to same space).
I nonneg. of H tends to produce part-based representationsbecause subtractive combinations are forbidden.
Early work by Paatero and Tapper (1994), landmark paper in Nature by Lee
and Seung (1999).
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 5: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/5.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
An unsupervised part-based representation
Along VQ, PCA or ICA, NMF provides an unsupervised linearrepresentation of data
vn ≈ W hn
data vectors “explanatory variables” “regressors”“basis”, “dictionary” “expansion coefficients”
“patterns” “activation coefficients”
and W is learnt from the set of data vectors V = [v1 . . . vN ].
I nonneg. of W ensures interpretability of the dictionary(features wk and data vn belong to same space).
I nonneg. of H tends to produce part-based representationsbecause subtractive combinations are forbidden.
Early work by Paatero and Tapper (1994), landmark paper in Nature by Lee
and Seung (1999).
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 6: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/6.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
49 images among 2429 from MIT’s CBCL face dataset
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 7: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/7.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
PCA dictionary with K = 25
red pixels indicate negative values
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 8: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/8.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
NMF dictionary with K = 25
as shown in (Lee and Seung, 1999)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 9: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/9.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
NMF as a constrained minimization problem
We seek to minimize a measure of fit between data V and modelWH, subject to nonnegativity of W and H:
minW,H≥0
D(V|WH) =∑fn
d([V]fn|[WH]fn)
where d(x |y) is a scalar cost function.
Regularization terms are often added to D(V|WH) to favorsparsity or smoothness of W or H.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 10: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/10.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Identifiability
Trivial scale and order ambiguities between the columns of W andthe rows of H.
There may be more complex geometrical ambiguities if the datadoes not fill the positive orthant “sufficiently well”.
I In the exact/noiseless case V = W?H?, the elements of thetrue dictionary W? need to belong to facets of the positiveorthant (Donoho and Stodden, 2004).
I In the approximate/noisy case, the situation is less clear, see(Laurberg, Christensen, Plumbley, Hansen, and Jensen, 2008).Adding regularization terms certainly help.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 11: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/11.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
β-divergence
A very popular cost function in NMF is the β-divergence (Basuet al., 1998; Eguchi and Kano, 2001; Cichocki and Amari, 2010),given by
dβ(x |y)def=
1
β (β−1)
(xβ + (β − 1) yβ − β x yβ−1
)β ∈ R\{0, 1}
x log xy + (y − x) β = 1
xy − log x
y − 1 β = 0
which takes the
I Euclidean distance (β = 2)
I Kullback-Leibler (KL) divergence (β = 1)
I Itakura-Saito (IS) divergence (β = 0)
as special cases.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 12: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/12.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
β-divergence
0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
d(x=1|y)
β = 2 (Euc)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 13: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/13.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
β-divergence
0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
d(x=1|y)
β = 2 (Euc)
β = 1 (KL)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 14: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/14.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
β-divergence
0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
d(x=1|y)
β = 2 (Euc)
β = 1 (KL)
β = 0 (IS)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 15: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/15.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
β-divergence
0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
d(x=1|y)
β = 2 (Euc)
β = 1 (KL)
β = 0 (IS)
β = −1
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 16: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/16.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
β-divergence
0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
d(x=1|y)
β = 2 (Euc)
β = 1 (KL)
β = 0 (IS)
β = −1
β = 3
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 17: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/17.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Common NMF algorithm design
I Block-coordinate update of H given W(i−1) and W given H(i)
minH≥0
D(V|W(i−1)H), minW≥0
D(V|WH(i))
I The updates of W and H are equivalent by symmetry:
V ≈WH ⇐⇒ VT ≈ HTWT
I The objective function is separable in the columns of H or therows of W:
D(V|WH) =∑n
D(vn|Whn)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 18: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/18.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Common NMF algorithm design
In the end we are left with
minh≥0
C (h)def= D(v|Wh)
which is a nonnegative linear regression problem that has receivedconsiderable attention in image restoration, e.g.,
I (Richardson, 1972; Lucy, 1974) with KL divergence
I (Daube-Witherspoon and Muehllehner, 1986; De Pierro,1993) for Euclidean distance
I (Cao, Eggermont, and Terebey, 1999) for Itakura-Saitodivergence (aka Burg entropy)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 19: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/19.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Majorization-minimization (MM)
Build G (h|h) such that G (h|h) ≥ C (h) and G (h|h) = C (h).Optimize (iteratively) G (h|h) instead of C (h).
0 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Objective function C(h)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 20: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/20.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Majorization-minimization (MM)
Build G (h|h) such that G (h|h) ≥ C (h) and G (h|h) = C (h).Optimize (iteratively) G (h|h) instead of C (h).
0 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
h(0)
h(1)
Objective function C(h)
Auxiliary function G(h|h(0)
)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 21: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/21.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Majorization-minimization (MM)
Build G (h|h) such that G (h|h) ≥ C (h) and G (h|h) = C (h).Optimize (iteratively) G (h|h) instead of C (h).
0 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
h(1)
h(2)
h(0)
Objective function C(h)
Auxiliary function G(h|h(1)
)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 22: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/22.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Majorization-minimization (MM)
Build G (h|h) such that G (h|h) ≥ C (h) and G (h|h) = C (h).Optimize (iteratively) G (h|h) instead of C (h).
0 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
h(3)
h(2)
h(1)
h(0)
Objective function C(h)
Auxiliary function G(h|h(2)
)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 23: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/23.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Majorization-minimization (MM)
Build G (h|h) such that G (h|h) ≥ C (h) and G (h|h) = C (h).Optimize (iteratively) G (h|h) instead of C (h).
0 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
h*
h(3)
h(2)
h(1)
h(0)
Objective function C(h)
Auxiliary function G(h|h*)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 24: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/24.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Majorization-minimization (MM)
Majorize convex and concave parts separately
C (h) =^
C (h)︸ ︷︷ ︸Maj. by Jensen’s ineq.
+_
C (h)︸ ︷︷ ︸Maj. by tangent
In the end, MM for β-NMF leads to
H←H.
[WT [(WH).(β−2).V]
WT [WH].(β−1)
]γ(β)
where γ(β) is a scalar β-dependent exponent, see (Nakano et al.,2010; Fevotte and Idier, 2011). Similar update for W.
Multiplicative form preserves nonnegativity given nonnegativeinitialization.Very easy to implement, of complexity O(FKN) per iteration.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 25: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/25.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Majorization-minimization (MM)
Majorize convex and concave parts separately
C (h) =^
C (h)︸ ︷︷ ︸Maj. by Jensen’s ineq.
+_
C (h)︸ ︷︷ ︸Maj. by tangent
In the end, MM for β-NMF leads to
H←H.
[WT [(WH).(β−2).V]
WT [WH].(β−1)
]γ(β)
where γ(β) is a scalar β-dependent exponent, see (Nakano et al.,2010; Fevotte and Idier, 2011). Similar update for W.
Multiplicative form preserves nonnegativity given nonnegativeinitialization.Very easy to implement, of complexity O(FKN) per iteration.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 26: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/26.jpg)
Generalities about NMF Itakura-Saito NMF Variants Concept of NMF Measure of fit Algorithms
Other algorithms
Specific to the Euclidean distance in most cases.
I Alternating nonnegative least squares (Paatero and Tapper,1994; Berry et al., 2007)
I Projected gradient descent (Lin, 2007)
I Interior-point gradient descent (Merritt and Zhang, 2005)
I Quasi-Newton methods (Zdunek and Cichocki, 2007; Kimet al., 2008)
I Active set methods (Kim and Park, 2008)
I Fast coordinate descent (Hsieh and Dhillon, 2011; Cichockiand Anh-Huy, 2009)
(selected references)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 27: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/27.jpg)
Generalities about NMF Itakura-Saito NMF Variants
Some applications of NMF...
I environmetrics (Paatero and Tapper, 1994)
I video summarization (Cooper and Foote, 2002)
I text mining (Lee and Seung, 1999; Xu et al., 2003)
I gene expression analysis (Brunet et al., 2004)
I Scotch whiskies clustering (Young et al., 2006) (!)
I hyperspectral imaging (Berry et al., 2007)
I portfolio diversification (Drakakis et al., 2007)
I clustering of protein interactions (Greene et al., 2008)
I food consumption analysis (Zetlaoui et al., 2010)
I image denoising and inpainting (Mairal et al., 2010)
(selected references)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 28: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/28.jpg)
Generalities about NMF Itakura-Saito NMF Variants
...and in particular
I music signal processing (Smaragdis and Brown, 2003)
activation coefficients
≈ ×
f
n
spectrogram patterns
WV H
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 29: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/29.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
Outline
Generalities about NMFConcept of NMFThe β-divergence: a unifying measure of fitMajorization-minimization algorithms
Itakura-Saito NMFStatistical modelPiano decomposition example
Variants of IS-NMFRegularized IS-NMFMultichannel IS-NMF
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 30: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/30.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
Model choices
activation coefficients
≈ ×
f
n
spectrogram patterns
WV H
I Magnitude or power spectrogram ?
I Which measure of fit should be used for the factorization ?
I NMF approximates the spectrogram by a sum of rank-onespectrograms. How can we invert these ? What about phase ?
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 31: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/31.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
Itakura-Saito NMF: a generative approach(Fevotte, Bertin, and Durrieu, 2009)
Let X = {xfn} be the (complex-valued) STFT of the signal.Assume
xfn =K∑
k=1
ck,fn
ck,fn ∼ Nc(0,wfkhkn)
and the components c1,fn, . . . , cK ,fn are independent given W andH.
Then
− log p(X|W,H) = DIS(|X|2|WH) + cst.
Additivity assumed in the STFT domain. Phase is preserved in themodel, though in a noninformative way (uniform distribution).
Related work by Benaroya et al. (2003); Parry and Essa (2007)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 32: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/32.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
Itakura-Saito NMF: a generative approach(Fevotte, Bertin, and Durrieu, 2009)
Let X = {xfn} be the (complex-valued) STFT of the signal.Assume
xfn =K∑
k=1
ck,fn
ck,fn ∼ Nc(0,wfkhkn)
and the components c1,fn, . . . , cK ,fn are independent given W andH. Then
− log p(X|W,H) = DIS(|X|2|WH) + cst.
Additivity assumed in the STFT domain. Phase is preserved in themodel, though in a noninformative way (uniform distribution).
Related work by Benaroya et al. (2003); Parry and Essa (2007)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 33: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/33.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
Itakura-Saito NMF: a generative approach(Fevotte, Bertin, and Durrieu, 2009)
Main message: Itakura-Saito NMF of the power spectrogramcorresponds to maximum likelihood estimation in a well-definedgenerative composite model of the STFT.
This in particular gives a statistically grounded way ofreconstructing the components:
ck,fn = E{ck,fn|X,W,H} =wfkhkn∑j wfjhjn︸ ︷︷ ︸
time-freq. mask
xfn
Lossless decomposition: xfn =∑
k ck,fn
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 34: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/34.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
Itakura-Saito NMF: a generative approach(Fevotte, Bertin, and Durrieu, 2009)
Alternatively, IS-NMF can be interpreted as maximum likelihood inmultiplicative noise:
vfn = |xfn|2 = [WH]fn . εfn
where εfn is exponential noise.
Related work by Abdallah and Plumbley (2004).
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 35: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/35.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
Noteworthy properties of the IS divergence
I The IS divergence is scale-invariant:
dIS(λ x |λ y) = dIS(x |y)
Implies higher accuracy in the representation of data withlarge dynamic range, such as audio spectra. In contrast,
dEUC (λ x |λ y) = λ2 dEUC (x |y)
dKL(λ x |λ y) = λ dKL(x |y)
I The IS divergence in nonconvex (inflexion at y = 2x); wasfound to lead to more local minima in practice.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 36: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/36.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
Other statistical factor models of the spectrogram
Latent factor models for count data inspired from text analysis:
I Poisson models (Le Roux et al., 2007; Cemgil, 2009), similarto GaP (Canny, 2004)
I Multinomial models (Shashanka et al., 2008; Smaragdis et al.,2009), similar to PLSI (Hofmann, 1999) or LDA (Blei et al.,2003; Buntine and Jakulin, 2006)
Not generative models:
I Data |xfn| is modeled as integer.
I Additivity is assumed at the magnitude level
|xfn| =∑
k|ck,fn|.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 37: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/37.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
Small-scale example
!!!! !!!! !!!! !!!!" ##### $(MIDI numbers : 61, 65, 68, 72)
Figure: Three representations of data.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 38: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/38.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
IS-NMF on power spectrogram with K = 8
−10−8−6−4−2
K =
1
Dictionary W
0
5000
10000
15000Coefficients H
−0.20
0.2
Reconstructed components
−10−8−6−4−2
K =
2
0
5000
10000
−0.20
0.2
−10−8−6−4−2
K =
3
0
2000
4000
6000
−0.20
0.2
−10−8−6−4−2
K =
4
02000400060008000
−0.20
0.2
−10−8−6−4−2
K =
5
0
1000
2000
−0.20
0.2
−10−8−6−4−2
K =
6
0
100
200
−0.20
0.2
−10−8−6−4−2
K =
7
0
2
4
−0.20
0.2
50 100 150 200 250 300 350 400 450 500−10−8−6−4−2
K =
8
0 100 200 300 400 500 6000
1
2
0.5 1 1.5 2 2.5 3x 105
−0.20
0.2
Pitch estimates: 65.0 68.0 61.0 72.0 0 0 0 0
(True values: 61, 65, 68, 72)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 39: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/39.jpg)
Generalities about NMF Itakura-Saito NMF Variants Statistical model Piano example
KL-NMF on magnitude spectrogram with K = 8
−6
−4
−2
K =
1
Dictionary W
0
50
100Coefficients H
−0.2
0
0.2
Reconstructed components
−6
−4
−2
K =
2
0
20
40
60
−0.2
0
0.2
−6
−4
−2
K =
3
0
20
40
−0.2
0
0.2
−6
−4
−2
K =
4
0
10
20
30
−0.2
0
0.2
−6
−4
−2
K =
5
0
20
40
−0.2
0
0.2
−6
−4
−2
K =
6
0
5
10
15
−0.2
0
0.2
−6
−4
−2
K =
7
0
5
10
−0.2
0
0.2
50 100 150 200 250 300 350 400 450 500
−6
−4
−2
K =
8
0 100 200 300 400 500 6000
2
4
0.5 1 1.5 2 2.5 3x 105
−0.2
0
0.2
Pitch estimates: 65.2 68.2 61.0 72.2 0 56.2 0 0
(True values: 61, 65, 68, 72)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 40: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/40.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Outline
Generalities about NMFConcept of NMFThe β-divergence: a unifying measure of fitMajorization-minimization algorithms
Itakura-Saito NMFStatistical modelPiano decomposition example
Variants of IS-NMFRegularized IS-NMFMultichannel IS-NMF
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 41: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/41.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Regularized NMF
One usually wants to reflect prior information/expectedcharacteristics about H or W in the form of penalty terms.
MM algorithms are easily adapted to penalized NMF, where a termR(W) or R(H) is added to the objective function.
Ex.: regularizer on H
D(V|WH)+R(H) ≤ G (H|H,W)+R(H)
Only the minimization step is changed. (Which may becomenon-tractable, in which case R(H) can be majorized itself.)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 42: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/42.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Regularized NMF
One usually wants to reflect prior information/expectedcharacteristics about H or W in the form of penalty terms.
MM algorithms are easily adapted to penalized NMF, where a termR(W) or R(H) is added to the objective function.
Ex.: regularizer on H
D(V|WH)+R(H) ≤ G (H|H,W)+R(H)
Only the minimization step is changed. (Which may becomenon-tractable, in which case R(H) can be majorized itself.)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 43: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/43.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMF
Audio exhibit time persistence/redundancy. Should be taken intoaccount in the factorization for
I more accurate estimation of H, and W,
I reduced identifiability ambiguities,
I perceptually more pleasant component reconstructions.
minW,H≥0
C (W,H) = DIS(V|WH) + λ
N∑n=2
D(hn|hn−1)
Choosing D(hn|hn−1) =∑
k dIS(hkn|hk,n−1) approximatelycorresponds to MAP estimation with an inverse-Gamma Markovchain prior on {hkn}n. See (Fevotte, 2011).
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 44: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/44.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMF
Audio exhibit time persistence/redundancy. Should be taken intoaccount in the factorization for
I more accurate estimation of H, and W,
I reduced identifiability ambiguities,
I perceptually more pleasant component reconstructions.
minW,H≥0
C (W,H) = DIS(V|WH) + λ
N∑n=2
D(hn|hn−1)
Choosing D(hn|hn−1) =∑
k dIS(hkn|hk,n−1) approximatelycorresponds to MAP estimation with an inverse-Gamma Markovchain prior on {hkn}n. See (Fevotte, 2011).
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 45: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/45.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFLouis Armstrong and His Hot Five
Log−power spectrogram
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Original data
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 46: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/46.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFEffect of regularization
Baseline (unpenalized IS−NMF)
Regularized (λ = 1)
Regularized (λ = 10)
3300 3400 3500 3600 3700 3800
Regularized (λ = 100)
Figure: Segment of one of the rows of H for different values of λ.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 47: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/47.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFComponent 1
Estimated Wiener filter 1
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Reconstructed component 1
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 48: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/48.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFComponent 2
Estimated Wiener filter 2
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Reconstructed component 2
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 49: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/49.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFComponent 3
Estimated Wiener filter 3
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Reconstructed component 3
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 50: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/50.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFComponent 4
Estimated Wiener filter 4
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Reconstructed component 4
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 51: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/51.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFComponent 5
Estimated Wiener filter 5
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Reconstructed component 5
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 52: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/52.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFComponent 6
Estimated Wiener filter 6
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Reconstructed component 6
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 53: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/53.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFComponent 7
Estimated Wiener filter 7
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Reconstructed component 7
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 54: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/54.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFComponent 8
Estimated Wiener filter 8
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Reconstructed component 8
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 55: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/55.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFComponent 9
Estimated Wiener filter 9
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Reconstructed component 9
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 56: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/56.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFComponent 10
Estimated Wiener filter 10
Fre
q.
20
40
60
80
100
120
10 20 30 40 50 60 70 80 90 100
−0.5
0
0.5
Reconstructed component 10
Am
p.
Time (s)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 57: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/57.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Smooth Itakura-Saito NMFFull audio restoration example
Original mono =Accompaniment︸ ︷︷ ︸
Comp. 1,9
+ Brass︸ ︷︷ ︸Comp. 2,3,5−8
+ Trombone︸ ︷︷ ︸Comp. 4
+ Noise︸ ︷︷ ︸Comp. 10
Original mono denoised
Original denoised & upmixed to stereo
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 58: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/58.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
IS-NMF with group sparsity(Lefevre, Bach, and Fevotte, 2011)
Expected activation structure in music:
H =
× × × × 0 0 × × × ×× × × × 0 0 × × × ×0 0 × × × × × × 0 00 0 × × × × × × 0 0× 0 × 0 × 0 × 0 × 0× 0 × 0 × 0 × 0 × 0
Exploit group structure to automate component grouping.
hn = [ × ×︸ ︷︷ ︸Source 1
× ×︸ ︷︷ ︸Source 2
× ×︸ ︷︷ ︸Source 3
]T
E.g., group sparsity: R(hn) = ‖h1,n‖2 + ‖h2,n‖2 + ‖h3,n‖2
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 59: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/59.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Automatic relevance determination in NMF(Tan and Fevotte, 2009, in press)
Tie each column wk and row hk through their prior, via a common“relevance” (scale) parameter: p(wk |Φk), p(hk |Φk).
+ ... +
φK
≈
V w1
h1
wK
hK
φ1
Some relevance parameters converge naturally towards a smallconstant and the corresponding components are pruned.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 60: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/60.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Automatic relevance determination in NMF(Tan and Fevotte, 2009, in press)
Under half-normal or exponential priors for wk , hk andinverse-Gamma prior for φk , MAP boils down to minimizing
C (W,H) = Dβ(V|WH) + λ
K∑k=1
log [f (wk) + f (hk) + b]
where
I f (x) = 12‖x‖2
2 or ‖x‖1
I b is a sparsity shape parameter
Concave term log(x + b) induces group-sparsity at the column &row level.Optimization can be handled in the MM framework.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 61: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/61.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Automatic relevance determination in NMFSwimmer data decomposition
(a) Noisy data
(b) `1-ARD decomposition wih K = 32
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 62: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/62.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Automatic relevance determination in NMFPiano data decomposition
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05
IS−NMF
0 5 10 15 200
0.01
0.02
0.03
0.04
0.05
ARD IS−NMF
Figure: Standard deviation of reconstructed components with IS-NMFand ARD IS-NMF applied to previously used piano data with K = 18.ARD IS-NMF retains the six “ground truth” components only.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 63: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/63.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
Multichannel IS-NMF(Ozerov and Fevotte, 2010)
+
+
Sources S NMF: W H Mixing system A
Mixture X
Multichannel NMF problem: Estimate W, H and A from X
noise 1
noise 2
Best scores on the underdetermined speech and music separation task at
the Signal Separation Evaluation Campaign (SiSEC) 2008.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 64: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/64.jpg)
Generalities about NMF Itakura-Saito NMF Variants Regularized IS-NMF Multichannel IS-NMF
User-guided multichannel IS-NMF(Ozerov, Fevotte, Blouet, and Durrieu, 2011)
I The decomposition is “guided” by the operator: sourceactivation time-codes are input to the separation system.
I The temporal segmentation is reflected in the form of zeros inH when a source is silent.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 65: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/65.jpg)
Generalities about NMF Itakura-Saito NMF Variants
Conclusions
I Itakura-Saito NMF of the power spectrogram is underlain by astatistical model of superimposed Gaussian components.
I This model is relevant to the representation of audio signals.
I Algorithms can be designed in a principled way in themajorization-minimization setting. Regularized variants.
I Possible extension to multichannel data for audio sourceseparation.
I Multiplicative updates make user-guided separation easy.
I The latent statistical model opens doors to fully Bayesianapproaches that integrates over W and/or H (Fevotte andCemgil, 2009; Hoffman et al., 2010; Fevotte et al., 2011;Dikmen and Fevotte, 2011)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 66: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/66.jpg)
Generalities about NMF Itakura-Saito NMF Variants
Conclusions
I Itakura-Saito NMF of the power spectrogram is underlain by astatistical model of superimposed Gaussian components.
I This model is relevant to the representation of audio signals.
I Algorithms can be designed in a principled way in themajorization-minimization setting. Regularized variants.
I Possible extension to multichannel data for audio sourceseparation.
I Multiplicative updates make user-guided separation easy.
I The latent statistical model opens doors to fully Bayesianapproaches that integrates over W and/or H (Fevotte andCemgil, 2009; Hoffman et al., 2010; Fevotte et al., 2011;Dikmen and Fevotte, 2011)
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 67: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/67.jpg)
Generalities about NMF Itakura-Saito NMF Variants
References I
S. A. Abdallah and M. D. Plumbley. Polyphonic transcription by nonnegative sparsecoding of power spectra. In Proc. 5th International Symposium Music InformationRetrieval (ISMIR), pages 318–325, Barcelona, Spain, Oct. 2004.
A. Basu, I. R. Harris, N. L. Hjort, and M. C. Jones. Robust and efficient estimation byminimising a density power divergence. Biometrika, 85(3):549–559, Sep. 1998.
L. Benaroya, R. Gribonval, and F. Bimbot. Non negative sparse representation forWiener based source separation with a single sensor. In Proc. IEEE InternationalConference on Acoustics, Speech and Signal Processing (ICASSP), pages 613–616,Hong Kong, 2003.
M. W. Berry, M. Browne, A. N. Langville, V. P. Pauca, and R. J. Plemmons.Algorithms and applications for approximate nonnegative matrix factorization.Computational Statistics & Data Analysis, 52(1):155–173, Sep. 2007.
D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent Dirichlet allocation. Journal ofMachine Learning Research, 3:993–1022, Jan. 2003.
J.-P. Brunet, P. Tamayo, T. R. Golub, and J. P. Mesirov. Metagenes and molecularpattern discovery using matrix factorization. In Proceedings of the NationalAcademy of Sciences, pages 4164–4169, Mar. 2004.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 68: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/68.jpg)
Generalities about NMF Itakura-Saito NMF Variants
References II
W. L. Buntine and A. Jakulin. Discrete component analysis. In Lecture Notes inComputer Science, volume 3940, pages 1–33. Springer, 2006. URLhttp://www.springerlink.com/content/d53027666542q3v7/.
J. F. Canny. GaP: A factor model for discrete data. In Proceedings of the 27th ACMinternational Conference on Research and Development of Information Retrieval(SIGIR), pages 122–129, 2004.
Y. Cao, P. P. B. Eggermont, and S. Terebey. Cross Burg entropy maximization and itsapplication to ringing suppression in image reconstruction. IEEE Transactions onImage Processing, 8(2):286–292, Feb. 1999. doi: 10.1109/83.743861.
A. T. Cemgil. Bayesian inference for nonnegative matrix factorisation models.Computational Intelligence and Neuroscience, 2009(Article ID 785152):17 pages,2009. doi:10.1155/2009/785152.
A. Cichocki and S. Amari. Families of Alpha- Beta- and Gamma- divergences: Flexibleand robust measures of similarities. Entropy, 12(6):1532–1568, June 2010.
A. Cichocki and P. Anh-Huy. Fast local algorithms for large scale nonnegative matrixand tensor factorizations. IEICE Transactions on fundamentals of electronics,communications and computer sciences, 92(3):708–721, 2009.
M. Cooper and J. Foote. Summarizing video using non-negative similarity matrixfactorization. In Proc. IEEE Workshop on Multimedia Signal Processing, 2002.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 69: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/69.jpg)
Generalities about NMF Itakura-Saito NMF Variants
References III
M. Daube-Witherspoon and G. Muehllehner. An iterative image space reconstructionalgorthm suitable for volume ECT. IEEE Transactions on Medical Imaging, 5(5):61– 66, 1986. doi: 10.1109/TMI.1986.4307748.
A. R. De Pierro. On the relation between the ISRA and the EM algorithm for positronemission tomography. IEEE Trans. Medical Imaging, 12(2):328–333, 1993. doi:10.1109/42.232263.
O. Dikmen and C. Fevotte. Nonnegative dictionary learning in the exponential noisemodel for adaptive music signal representation. In J. Shawe-Taylor, R. Zemel,P. Bartlett, F. Pereira, and K. Weinberger, editors, Advances in Neural InformationProcessing Systems 24 (NIPS), pages 2267–2275, Granada, Spain, Dec. 2011. MITPress. URLhttp://perso.telecom-paristech.fr/~fevotte/Proceedings/nips11.pdf.
D. Donoho and V. Stodden. When does non-negative matrix factorization give acorrect decomposition into parts? In S. Thrun, L. Saul, and B. Scholkopf, editors,Advances in Neural Information Processing Systems 16. MIT Press, Cambridge,MA, 2004.
K. Drakakis, S. Rickard, R. de Frein, and A. Cichocki. Analysis of financial data usingnon-negative matrix factorization. International Journal of Mathematical Sciences,6(2), June 2007.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 70: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/70.jpg)
Generalities about NMF Itakura-Saito NMF Variants
References IV
S. Eguchi and Y. Kano. Robustifying maximum likelihood estimation. Technicalreport, Institute of Statistical Mathematics, June 2001. URLhttp://www.ism.ac.jp/$\sim$eguchi/pdf/Robustify_MLE.pdf. ResearchMemo. 802.
C. Fevotte. Majorization-minimization algorithm for smooth Itakura-Saito nonnegativematrix factorization. In Proc. International Conference on Acoustics, Speech andSignal Processing (ICASSP), Prague, Czech Republic, May 2011. URLhttp://perso.telecom-paristech.fr/~fevotte/Proceedings/icassp11a.pdf.
C. Fevotte and A. T. Cemgil. Nonnegative matrix factorisations as probabilisticinference in composite models. In Proc. 17th European Signal ProcessingConference (EUSIPCO), pages 1913–1917, Glasgow, Scotland, Aug. 2009. URLhttp://www.tsi.enst.fr/~fevotte/Proceedings/eusipco09_1.pdf.
C. Fevotte and J. Idier. Algorithms for nonnegative matrix factorization with thebeta-divergence. Neural Computation, 23(9):2421–2456, Sep. 2011. doi:10.1162/NECO a 00168. URL http://arxiv.org/pdf/1010.1763v3.
C. Fevotte, N. Bertin, and J.-L. Durrieu. Nonnegative matrix factorization with theItakura-Saito divergence. With application to music analysis. Neural Computation,21(3):793–830, Mar. 2009. doi: 10.1162/neco.2008.04-08-771. URLhttp://www.tsi.enst.fr/~fevotte/Journals/neco09_is-nmf.pdf.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 71: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/71.jpg)
Generalities about NMF Itakura-Saito NMF Variants
References V
C. Fevotte, O. Cappe, and A. T. Cemgil. Efficient Markov chain Monte Carloinference in composite models with space alternating data augmentation. InProc. IEEE Workshop on Statistical Signal Processing (SSP), pages 221 – 224,Nice, France, June 2011. URLhttp://perso.telecom-paristech.fr/~fevotte/Proceedings/ssp11.pdf.
D. Greene, G. Cagney, N. Krogan, and P. Cunningham. Ensemble non-negative matrixfactorization methods for clustering protein-protein interactions. Bioinformatics, 24(15):1722–1728, 2008.
M. Hoffman, D. Blei, and P. Cook. Bayesian nonparametric matrix factorization forrecorded music. In Proc. 27th International Conference on Machine Learning(ICML), Haifa, Israel, 2010.
T. Hofmann. Probabilistic latent semantic indexing. In Proc. 22nd InternationalConference on Research and Development in Information Retrieval (SIGIR), 1999.URL http://www.cs.brown.edu/~th/papers/Hofmann-SIGIR99.pdf.
C. J. Hsieh and I. S. Dhillon. Fast coordinate descent methods with variable selectionfor non-negative matrix factorization. In Proc. 17th ACM SIGKDD InternationalConference on Knowledge Discovery and Data Mining (KDD), pages 1064 – 1072,Aug. 2011.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 72: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/72.jpg)
Generalities about NMF Itakura-Saito NMF Variants
References VI
D. Kim, S. Sra, and I. S. Dhillon. Fast projection-based methods for the least squaresnonnegative matrix approximation problem. Statistical Analysis and Data Mining, 1(1):38–51, 2008.
H. Kim and H. Park. Nonnegative matrix factorization based on alternatingnonnegativity constrained least squares and active set method. SIAM Journal onMatrix Analysis and Applications, 30(2):713–730, 2008.
H. Laurberg, M. Christensen, M. D. Plumbley, L. K. Hansen, and S. H. Jensen.Theorems on positive data: On the uniqueness of NMF. Computational Intelligenceand Neuroscience, Article ID 764206, 2008.
J. Le Roux, H. Kameoka, N. Ono, A. de Cheveigne, and S. Sagayama. Single andmultiple F0 contour estimation through parametric spectrogram modeling ofspeech in noisy environments. IEEE Transactions on Audio, Speech and LanguageProcessing, 15(4):1135–1145, May 2007.
D. D. Lee and H. S. Seung. Learning the parts of objects with nonnegative matrixfactorization. Nature, 401:788–791, 1999.
A. Lefevre, F. Bach, and C. Fevotte. Itakura-Saito nonnegative matrix factorizationwith group sparsity. In Proc. International Conference on Acoustics, Speech andSignal Processing (ICASSP), Prague, Czech Republic, May 2011. URLhttp://perso.telecom-paristech.fr/~fevotte/Proceedings/icassp11c.pdf.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 73: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/73.jpg)
Generalities about NMF Itakura-Saito NMF Variants
References VII
C.-J. Lin. Projected gradient methods for nonnegative matrix factorization. NeuralComputation, 19:2756–2779, 2007.
L. B. Lucy. An iterative technique for the rectification of observed distributions.Astronomical Journal, 79:745–754, 1974. doi: 10.1086/111605.
J. Mairal, F. Bach, J. Ponce, and G. Sapiro. Online learning for matrix factorizationand sparse coding. Journal of Machine Learning Research, 11:10–60, 2010.
M. Merritt and Y. Zhang. Interior-point gradient method for large-scale totallynonnegative least squares problems. Journal of optimization theory andapplications, 126(1):191–202, 2005.
M. Nakano, H. Kameoka, J. Le Roux, Y. Kitano, N. Ono, and S. Sagayama.Convergence-guaranteed multiplicative algorithms for non-negative matrixfactorization with beta-divergence. In Proc. IEEE International Workshop onMachine Learning for Signal Processing (MLSP’2010), Sep. 2010.
A. Ozerov and C. Fevotte. Multichannel nonnegative matrix factorization inconvolutive mixtures for audio source separation. IEEE Transactions on Audio,Speech and Language Processing, 18(3):550–563, Mar. 2010. doi:10.1109/TASL.2009.2031510. URLhttp://www.tsi.enst.fr/~fevotte/Journals/ieee_asl_multinmf.pdf.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 74: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/74.jpg)
Generalities about NMF Itakura-Saito NMF Variants
References VIII
A. Ozerov, C. Fevotte, R. Blouet, and J.-L. Durrieu. Multichannel nonnegative tensorfactorization with structured constraints for user-guided audio source separation. InProc. International Conference on Acoustics, Speech and Signal Processing(ICASSP), Prague, Czech Republic, May 2011. URLhttp://perso.telecom-paristech.fr/~fevotte/Proceedings/icassp11d.pdf.
P. Paatero and U. Tapper. Positive matrix factorization : A non-negative factor modelwith optimal utilization of error estimates of data values. Environmetrics, 5:111–126, 1994.
R. M. Parry and I. Essa. Phase-aware non-negative spectrogram factorization. InProc. International Conference on Independent Component Analysis and SignalSeparation (ICA), pages 536–543, London, UK, Sep. 2007.
W. H. Richardson. Bayesian-based iterative method of image restoration. Journal ofthe Optical Society of America, 62:55–59, 1972.
M. Shashanka, B. Raj, and P. Smaragdis. Probabilistic latent variable models asnonnegative factorizations. Computational Intelligence and Neuroscience, 2008(Article ID 947438):8 pages, 2008. doi:10.1155/2008/947438.
P. Smaragdis and J. C. Brown. Non-negative matrix factorization for polyphonic musictranscription. In IEEE Workshop on Applications of Signal Processing to Audio andAcoustics (WASPAA’03), Oct. 2003.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 75: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/75.jpg)
Generalities about NMF Itakura-Saito NMF Variants
References IX
P. Smaragdis, M. Shashanka, and B. Raj. A sparse non-parametric approach for singlechannel separation of known sounds. In Y. Bengio, D. Schuurmans, J. Lafferty,C. K. I. Williams, and A. Culotta, editors, Advances in Neural InformationProcessing Systems 22, pages 1705–1713. MIT Press, 2009.
V. Y. F. Tan and C. Fevotte. Automatic relevance determination in nonnegativematrix factorization. In Proc. Workshop on Signal Processing with AdaptativeSparse Structured Representations (SPARS), St-Malo, France, Apr. 2009. URLhttp://www.tsi.enst.fr/~fevotte/Proceedings/spars09.pdf.
V. Y. F. Tan and C. Fevotte. Automatic relevance determination in nonnegative matrixfactorization with the beta-divergence. IEEE Transactions on Pattern Analysis andMachine Intelligence, in press. URL http://arxiv.org/abs/1111.6085.
W. Xu, X. Liu, and Y. Gong. Document clustering based on non-negative matrixfactorization. In Proc. 26th International ACM SIGIR conference on research anddevelopment in information retrieval (SIGIR’03), pages 267–273, Toronto, Canada,2003.
S. S. Young, P. Fogel, and D. Hawkins. Clustering Scotch whiskies using non-negativematrix factorization. Joint Newsletter for the Section on Physical and EngineeringSciences and the Quality and Productivity Section of the American StatisticalAssociation, 14(1):11–13, June 2006.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization
![Page 76: Itakura-Saito nonnegative matrix factorization and friends ... · Generalities about NMFItakura-Saito NMFVariants Itakura-Saito nonnegative matrix factorization and friends for music](https://reader034.vdocument.in/reader034/viewer/2022042023/5e7bd257826f46143a34d766/html5/thumbnails/76.jpg)
Generalities about NMF Itakura-Saito NMF Variants
References X
R. Zdunek and A. Cichocki. Nonnegative matrix factorization with constrainedsecond-order optimization. Signal Processing, 87(8):1904–1916, Aug. 2007.
M. Zetlaoui, M. Feinberg, P. Verger, and S. Clemencon. Extraction of foodconsumption systems by non-negative matrix factorization (nmf) for theassessment of food choices. Technical report, Arxiv, 2010. URLhttp://hal.archives-ouvertes.fr/docs/00/48/47/94/PDF/NMF_food.pdf.
Cedric Fevotte (CNRS) Itakura-Saito nonnegative matrix factorization