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Single-channel source separation using non-negative matrix factorization
Mikkel N. SchmidtTechnical University of [email protected]
DTU InformaticsDepartment of Informatics and Mathematical Modeling
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Agenda
Single-channel source separation
Non-negative matrix factorization
NMF 2-D deconvolution (with Morten Mørup)
Speech separation using sparse NMF (with Rasmus K. Olsson)
NMF with Gaussian process priors (with Hans Laurberg)
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Single-channel source separation
Additive model
Mixture Sources Noise
Under-determined problem: More information required
Example: Two-source noise-free
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Approaches to single-channel source separation
Filtering
Decomposition and grouping
Source modelling
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Model-based source-separation
model
reconstruction
featureextraction
featureextraction
separation
trainingdata
mixture
estimate
data
infe
renc
e
repr
esen
tatio
n
mod
el
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Signal representation
Emphasize desired characteristics
Introduce invariances
Allow assumptions of independence or exchangeability
Reduce dimensionality
Allow signal reconstruction
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Model
Mixing model
Source model
Model building
Model training
Model adaptation
Goals
Accurately model sources and mixing process
Enable efficient inference
Likelihood
Prior
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Inference
Estimate sources:
Maximum a posteriori, posterior mean, marginal MAP, etc.
Solve optimization or integration problem
Posterior Likelihood Prior
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Non-negative matrix factorization
Data Factorizing matrices
Non-negativity constraints
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Non-negative matrix factorization
Non-negative bilinear model
Sum of products of non-negative variables
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Why non-negativity?
Many signals are naturally non-negative
Pixel intensities
Amplitude spectra
Occurrence counts
Discrete probabilities
Additive combination of features
No cancellations
Build the whole as a sum of parts
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Basic computation
Divergence measure
Constraints
Constrained minimization problem
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Probabilistic formulation
Likelihood Prior
Maximum likelihood
Maximum a posteriori
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Divergence measures (Likelihoods)
Squared error (Lee and Seung, 1999)
Kullback-Leibler divergence (Lee and Seung, 1999)
Bregman's divergence (Dhillon and Sra, 2005)
Kompass' divergence (Kompass, 2007)
Csiszár's divergence (Cichocki et al., 2006)
Amari's alpha divergence (Cichocki et al., 2006)
Weighted versions of the above (Guillamet et al., 2001)
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Distribution of the factors (Priors)
Sparsity (Hoyer, 2002)
Orthogonality (Ding et al., 2005)
Discriminative (Wang et al., 2004)
Smoothness (Virtanen, 2003)
Gaussian process (Schmidt and Laurberg, 2008)
Transformation invariance (Wersing et al., 2003)
Convolutive (Virtanen, 2004; Smaragdis, 2004)
2-D convolutive (Schmidt and Mørup, 2006)
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Optimization strategies
Direct optimization
Alternating optimization
Alternating descent
Can haveconvex
subproblems
Non-convex
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Algorithms
Projected least squares (Paatero, 1997)
Multiplicative updates (Lee and Seung, 1999)
Projected gradient descent (Lin, 2007)
Logarithmic barrier method (Lu and Wu, 2005)
Active set (Berry et al., 2006)
Quasi Newton (Kim et al., 2007)
Reparametrization (Cichocki et al., 2006)
SOCP (Heiler and Schnörr, 2006)
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NMF 2-D deconvolution
NMF
NMF 2-D
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Example: Flute and guitar
Time [seconds]
Freq
uenc
y [k
Hz]
2 4 6 8 10
.2
.4
.8
1.6
3.2
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Speech separation using sparse NMF
Learn basis for each source
Basis
Training data
Mixture
Decompose mixture
Reconstruct each source
Source estimate
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Example: Two speakers
Time [seconds]1 2 3 4
Freq
uenc
y [k
Hz]
3.5
3
2.5
2
1.5
1
.5
MixtureBases
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NMF with Gaussian process priors
Data Gaussian processes
Link functions
GP: General distribution over functions
Link function: Non-linear map to non-negative reals
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Gaussian process
Distribution over functions
Characterized by
Mean function
Covariance function
Example
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Change of variable
Cholesky decomposition
I.I.D. standard normal
Same model, different parametrization
Parameters are a priori uncorrelated
Empirically better optimization properties
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Illustration of NMF with GP priors
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Example: Toy problem
Noisy data Underlying data
NMF GPP-NMF
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Example: Toy problem
Columns of A Rows of B
Underlying data
NMF
GPP-NMF
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Example: Chemical shift brain imaging
Data: 369 point spectra @ 512 points (888 grid) in human head
Task: Distinguish between brain and muscle tissue
Spectra prior: Smooth exponential distribution
Activation prior: 3-D smooth, exponential distribution, left-to-right symmetric
Random draw from prior
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Example: Chemical shift brain imaging
Non-negative matrix factorization
NMF with Gaussianprocess priors
Mus
cle
Bra
in
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Conclusions
Single-channel source separation is a difficult problem that occurs in many areas
Model based source separation provides a principled approach to solving the problem
NMF and its extensions are useful models for single-channel source separation
Gaussian processes can be used as a general framework for incorporating prior information in NMF
