Bayesian Nonparametric Matrix Factorization for Recorded Music

Matthew D. Hoffman, David M. Blei, Perry R. Cook

Presented by Lu Ren

Electrical and Computer Engineering

Duke University

Outline

Introduction

GaP-NMF Model

Variational Inference

Evaluation

Related Work

Conclusions

Introduction

Breaking audio spectrograms into separate sources of sound

Identifying individual instruments and notes

Predicting hidden or distorted signals

Source separation

previous work

Specifying the number of sources---Bayesian Nonparametric Gamma Process Nonnegative Matrix Factorization (GaP-NMF) 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 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

Assume K static sound sources

: describe these sources

: amplitude of each source changing over time

is the average amount of energy source k exhibits at frequency m

is the gain of source k at time n

GaP-NMF Model

1Abdallah & Plumbley (2004) and Fevotte et al. (2009)

Mixing K sound sources in the time domain (under certain assumptions), spectrogram is distributed1

Infer both the characters and number of latent audio sources

: trunction level

GaP-NMF Model

As goes infinity, approximates an infinite sequence 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

are substantially greater than 0. Setting :

θ

θ

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 Tightening the likelihood bound

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

Evaluation

1. Synthetic Data

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