Bayesian Inference for Nonnegative Matrix Factorisation Models

We describe nonnegative matrix factorisation (NMF) with a Kullback-Leibler (KL) error measure in a statistical framework, with a hierarchical generative model consisting of an observation and a prior component. Omitting the prior leads to the standard KL-NMF algorithms as special cases, where maximum likelihood parameter estimation is carried out via the Expectation-Maximisation (EM) algorithm. Starting from this view, we develop full Bayesian inference via variational Bayes or Monte Carlo. Our construction retains conjugacy and enables us to develop more powerful models while retaining attractive features of standard NMF such as monotonic convergence and easy implementation. We illustrate our approach on model order selection and image reconstruction

Multichannel high resolution NMF for modelling convolutive mixtures of non-stationary signals in the time-frequency domain

Several probabilistic models involving latent components have been proposed for modeling time-frequency (TF) representations of audio signals such as spectrograms, notably in the nonnegative matrix factorization (NMF) literature. Among them, the recent high-resolution NMF (HR-NMF) model is able to take both phases and local correlations in each frequency band into account, and its potential has been illustrated in applications such as source separation and audio inpainting. In this paper, HR-NMF is extended to multichannel signals and to convolutive mixtures. The new model can represent a variety of stationary and non-stationary signals, including autoregressive moving average (ARMA) processes and mixtures of damped sinusoids. A fast variational expectation-maximization (EM) algorithm is proposed to estimate the enhanced model. This algorithm is applied to piano signals, and proves capable of accurately modeling reverberation, restoring missing observations, and separating pure tones with close frequencies

An Oracle Inequality for Quasi-Bayesian Non-Negative Matrix Factorization

The aim of this paper is to provide some theoretical understanding of quasi-Bayesian aggregation methods non-negative matrix factorization. We derive an oracle inequality for an aggregated estimator. This result holds for a very general class of prior distributions and shows how the prior affects the rate of convergence.Comment: This is the corrected version of the published paper P. Alquier, B. Guedj, An Oracle Inequality for Quasi-Bayesian Non-negative Matrix Factorization, Mathematical Methods of Statistics, 2017, vol. 26, no. 1, pp. 55-67. Since then Arnak Dalalyan (ENSAE) found a mistake in the proofs. We fixed the mistake at the price of a slightly different logarithmic term in the boun

Link Prediction via Generalized Coupled Tensor Factorisation

This study deals with the missing link prediction problem: the problem of predicting the existence of missing connections between entities of interest. We address link prediction using coupled analysis of relational datasets represented as heterogeneous data, i.e., datasets in the form of matrices and higher-order tensors. We propose to use an approach based on probabilistic interpretation of tensor factorisation models, i.e., Generalised Coupled Tensor Factorisation, which can simultaneously fit a large class of tensor models to higher-order tensors/matrices with com- mon latent factors using different loss functions. Numerical experiments demonstrate that joint analysis of data from multiple sources via coupled factorisation improves the link prediction performance and the selection of right loss function and tensor model is crucial for accurately predicting missing links