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 Variational Inference Evaluation Related Work Conclusions Introduction GaP-NMF Model Variational Inference Evaluation Related Work Conclusions

Introduction Specifying the number of sources---Bayesian Nonparametric 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 : M by N matrix of nonnegative reals 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