Ali Taylan Cemgil

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.

On tempo tracking: Tempogram representation and Kalman filtering

We formulate tempo tracking in a Bayesian framework where a tempo tracker is modeled as a stochastic dynamical system. The tempo is modeled as a hidden state variable of the system and is estimated by a Kalman filter. The Kalman filter operates on a Tempogram, a wavelet-like multiscale expansion of a real performance. An important advantage of our approach is that it is possible to formulate both off-line or real-time algorithms. The simulation results on a systematically collected set of MIDI piano performances of Yesterday and Michelle by the Beatles shows accurate tracking of approximately 90% of the beats.

A generative model for music transcription

In this paper, we present a graphical model for polyphonic music transcription. Our model, formulated as a dynamical Bayesian network, embodies a transparent and computationally tractable approach to this acoustic analysis problem. An advantage of our approach is that it places emphasis on explicitly modeling the sound generation procedure. It provides a clear framework in which both high level (cognitive) prior information on music structure can be coupled with low level (acoustic physical) information in a principled manner to perform the analysis. The model is a special case of the, generally intractable, switching Kalman filter model. Where possible, we derive, exact polynomial time inference procedures, and otherwise efficient approximations. We argue that our generative model based approach is computationally feasible for many music applications and is readily extensible to more general auditory scene analysis scenarios.

Monte Carlo methods for tempo tracking and rhythm quantization

We present a probabilistic generative model for timing deviations in expressive music performance. The structure of the proposed model is equivalent to a switching state space model. The switch variables correspond to discrete note locations as in a musical score. The continuous hidden variables denote the tempo. We formulate two well known music recognition problems, namely tempo tracking and automatic transcription (rhythm quantization) as filtering and maximum a posteriori (MAP) state estimation tasks. Exact computation of posterior features such as the MAP state is intractable in this model class, so we introduce Monte Carlo methods for integration and optimization. We compare Markov Chain Monte Carlo (MCMC) methods (such as Gibbs sampling, simulated annealing and iterative improvement) and sequential Monte Carlo methods (particle filters). Our simulation results suggest better results with sequential methods. The methods can be applied in both online and batch scenarios such as tempo tracking and transcription and are thus potentially useful in a number of music applications such as adaptive automatic accompaniment, score typesetting and music information retrieval.

Bayesian extensions to non-negative matrix factorisation for audio signal modelling

We describe the underlying probabilistic generative signal model of non-negative matrix factorisation (NMF) and propose a realistic conjugate priors on the matrices to be estimated. A conjugate Gamma chain prior enables modelling the spectral smoothness of natural sounds in general, and other prior knowledge about the spectra of the sounds can be used without resorting to too restrictive techniques where some of the parameters are fixed. The resulting algorithm, while retaining the attractive features of standard NMF such as fast convergence and easy implementation, outperforms existing NMF strategies in a single channel audio source separation and detection task.

Rhythm Quantization for Transcription

Automatic Music Transcription is the extraction of an acceptable notation from performed music. One important task in this problem is rhythm quantization which refers to categorization of note durations. Although quantization of a pure mechanical performance is rather straightforward, the task becomes increasingly difficult in presence of musical expression, i.e. systematic variations in timing of notes and tempo changes. For quantization of natural performances, we employ a framework based on Bayesian statistics. We demonstrate that some simple quantization schemata can be derived in this framework by simple assumptions about timing deviations. A general quantization method, which can be derived in this framework, is vector quantization (VQ). The algorithm operates on short groups of onsets and is thus flexible in capturing the structure of timing deviations between neighboring onsets and thus performs better than simple rounding methods. Finally, we present some results on simple examples.

Mixtures of Gamma Priors for Non-negative Matrix Factorization Based Speech Separation

This paper deals with audio source separation using supervised non-negative matrix factorization (NMF). We propose a prior model based on mixtures of Gamma distributions for each sound class, which hyperparameters are trained given a training corpus. This formulation allows adapting the spectral basis vectors of the sound sources during actual operation, when the exact characteristics of the sources are not known in advance. Simulations were conducted using a random mixture of two speakers. Even without adaptation the mixture model outperformed the basic NMF, and adaptation furher improved slightly the separation quality. Audio demonstrations are available at www.cs.tut.fi/~tuomasv .

Gamma Markov Random Fields for Audio Source Modeling

In many audio processing tasks, such as source separation, denoising or compression, it is crucial to construct realistic and flexible models to capture the physical properties of audio signals. This can be accomplished in the Bayesian framework through the use of appropriate prior distributions. In this paper, we describe a class of prior models called Gamma Markov random fields (GMRFs) to model the sparsity and the local dependency of the energies (i.e., variances) of time-frequency expansion coefficients. A GMRF model describes a non-normalised joint distribution over unobserved variance variables, where given the field the actual source coefficients are independent. Our construction ensures a positive coupling between the variance variables, so that signal energy changes smoothly over both axes to capture the temporal and spectral continuity. The coupling strength is controlled by a set of hyperparameters. Inference on the overall model is convenient because of the conditional conjugacy of all of the variables in the model, but automatic optimization of hyperparameters is crucial to obtain better fits. The marginal likelihood of the model is not available because of the intractable normalizing constant of GMRFs. In this paper, we optimize the hyperparameters of our GMRF-based audio model using contrastive divergence and compare this method to alternatives such as score matching and pseudolikelihood maximization where applicable. We present the performance of the GMRF models in denoising and single-channel source separation problems in completely blind scenarios, where all the hyperparameters are jointly estimated given only audio data.

Generative model based polyphonic music transcription

We present a model for simultaneous tempo and polyphonic pitch tracking. Our model, a form of dynamic Bayesian network (Murphy, K.P., 2002), embodies a transparent and computationally tractable approach to this acoustic analysis problem. An advantage of our approach is that it places emphasis on modeling the sound generation procedure. It provides a clear framework in which both high level (cognitive) prior information on music structure can be coupled with low level (acoustic physical) information in a principled manner to perform the analysis. The model is readily extensible to more complex sound generation processes.

Approximate Bayesian methods for kernel-based object tracking

A framework for real-time tracking of complex non-rigid objects is presented. The object shape is approximated by an ellipse and its appearance by histogram based features derived from local image properties. An efficient search procedure is used to find the image region with a histogram most similar to the histogram of the tracked object. The procedure is a natural extension of the mean-shift procedure with Gaussian kernel which allows handling the scale and orientation changes of the object. The presented procedure is integrated into a set of Bayesian filtering schemes. We compare the regular and mixture Kalman filter and other sequential importance sampling (particle filtering) techniques.