Matthew D. Hoffman, David M. Blei, Perry R. Cook
Presented by Lu Ren
Electrical and Computer Engineering
Duke University

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Outline

Introduction
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Breaking audio spectrograms into separate sources of sound previous work
Identifying individual instruments and notes
Predicting hidden or distorted signals
Source separation
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Specifying the number of sources---Bayesian Nonparametric
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Gamma Process Nonnegative Matrix Factorization (GaP-NMF)
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Computational challenge: non-conjugate pairs of distributions
• favor for spectrogram data, not for computational convenience
• bigger variational family analytic coordinate ascent algorithm

GaP-NMF Model
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Observation: Fourier power sepctrogram of an audio signal
: M by N matrix of nonnegative reals
: power at time window n and frequency bin m
A window of
2(M-1) samples
DFT
Squared magnitude in each frequency bin
Keep only the first M bins
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Assume K static sound sources
: describe these sources is the average amount of energy source k exhibits at frequency m
: amplitude of each source changing over time is the gain of source k at time n

GaP-NMF Model
Mixing K sound sources in the time domain (under certain assumptions), spectrogram is distributed 1
Infer both the characters and number of latent audio sources
: trunction level
1 Abdallah & Plumbley (2004) and Fevotte et al. (2009)

GaP-NMF Model
 drawn from a gamma process
 Number of elements greater than some is finite almost surely:
 If is sufficiently large relative to , only a few elements of
θ are substantially greater than 0.
 Setting :

Variational Inference
Variational distribution: expanded family
Generalized Inverse-Gaussian (GIG): denotes a modified Bessel function of the second kind
Gamma family is a special case of the GIG family where ,

Variational Inference
Lower bound of GaP-NMF model:
If :
GIG family sufficient statistics:
Gamma family sufficient statistics:

Variational Inference
The likelihood term expands to:
With Jensen’s inequality:

Variational Inference
With a first order Taylor approximation:
: an arbitrary positive point

Variational Inference
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Tightening the likelihood bound
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Optimizing the variational distributions
For example:

Evaluation
Compare GaP-NMF to two variations:
1. Finite Bayesian model
2. Finite non-Bayesian model
Itakura-Saito Nonnegative Matrix Factorization (IS-NMF)
: maximize the likelihood in the above fomula
Compare with another two NMF algorithms:
EU-NMF: minimize the sum of the squared Euclidean distance
KL-NMF: minimize the generalized KL-divergence

1. Synthetic Data
Evaluation

Evaluation
2. Marginal Likelihood & Bandwidth Expansion

Evaluation
3. Blind Monophonic Source Separation

Conclusions
 Related work
 Bayesian nonparametric model GaP-NMF
 Applicable to other types of audio