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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 Squared magnitude in each frequency bin DFT Keep only the first M bins • 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 distributed1 Infer both the characters and number of latent audio sources : trunction level 1Abdallah & Plumbley (2004) and Fevotte et al. (2009)
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 :
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