Bayesian Nonparametric Matrix Factorization for Recorded Music Reading Group Presenter: Shujie Hou Cognitive Radio Institute Friday, October 15, 2010 Authors: Matthew D. Hoffman, David M. Blei, Perry R. cook Princeton University, Department of Computer Science, 35 olden St., Princeton, NJ, 08540 USA
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Bayesian Nonparametric Matrix Factorization for Recorded Music Reading Group Presenter: Shujie Hou Cognitive Radio Institute Friday, October 15, 2010 Authors:
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Bayesian Nonparametric Matrix Factorization for Recorded Music
Reading Group Presenter:
Shujie Hou
Cognitive Radio Institute
Friday, October 15, 2010
Authors: Matthew D. Hoffman, David M. Blei, Perry R. cook
Princeton University, Department of Computer Science, 35 olden St., Princeton, NJ, 08540 USA
Outline
■ Introduction■ Terminology■ Problem statement and contribution of this paper
■ Gap-NMF Model(Gamma Process Nonnegative Matrix Factorization )
■ Nonparametric Statistics:□ The term non-parametric is not meant to imply that such models
completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance.
■ Nonnegative Matrix Factorization:□ Non-negative matrix factorization (NMF) is a group of algorithms
in multivariate analysis and linear algebra where a matrix, is factorized into (usually) two matrices with all elements are greater than or equal to 0 WHX
The above two definitions are cited from Wikipedia
Terminology(2)
■ Variational Inference:□ Variational inference approximates the posterior distribution with
a simpler distribution, whose parameters are optimized to be close to the true posterior.
■ Mean-field Variational Inference:□ In mean-field variational inference, each variable is given an
independent distribution, usually of the same family as its prior.
Outline
■ Introduction■ Terminology■ Problem statement and Contribution of this Paper
in which is the power of audio signal at time window n and frequency bin m.
If the number of latent variable is specified in advance:■ Assuming the audio signal is composed of K static sound
sources. The problem is to decompose , in which is M by K matrix, is K by N matrix. In which cell is the average amount of energy source k exhibits at frequency m. cell is the gain of source k at time n.
■ The problem is solved by
X
mnX
WHX
H
W
knH
mkW
GaP-NMF Model
If the number of latent variable is not specified in advance:
■ GaP-NMF assumes that the data is drawn according to the following generative process:
Based on the formula that(Abdallah&Plumbley (2004))
GaP-NMF Model
If the number of latent variable is not specified in advance:
■ GaP-NMF assumes that the data is drawn according to the following generative process:
The overall gain of the corresponding source l
Based on the formula that(Abdallah&Plumbley (2004))
Used to control the number of latent variables
GaP-NMF Model
■ The number of nonzero is the number of the latent variables K.
■ If L increased towards infinity, the nonzero L which expressed by K is finite and obeys:
Kingman ,1993
Outline
■ Introduction■ Terminology■ Problem statement and Contribution of this Paper
■ Variational inference approximates the posterior distribution with a simpler distribution, whose parameters are optimized to be close to the true posterior.
■ Under this paper’s condition:
Posterior Distribution
What measured
Definition of Variational Inference
■ Variational inference approximates the posterior distribution with a simpler distribution, whose parameters are optimized to be close to the true posterior.
■ Under this paper’s condition:
Variational distribution assumption with free parameters
Variational Distribution Posterior Distribution
Approximates
What measured
Definition of Variational Inference
■ Variational inference approximates the posterior distribution with a simpler distribution, whose parameters are optimized to be close to the true posterior.
■ Under this paper’s condition:
Variational distribution assumption with free parameters
Variational Distribution Adjust Parameters Posterior Distribution
Approximates
What measured
Outline
■ Introduction■ Terminology■ Problem statement and Contribution of this Paper