Cédric Févotte

• PhD
• Researcher at French National Centre for Scientific Research

This article introduces new multiplicative updates for nonnegative matrix factorization with the $\beta$ -divergence and sparse regularization of one of the two factors (say, the activation matrix). It is well known that the norm of the other factor (the dictionary matrix) needs to be controlled in order to avoid an ill-posed formulation. Standar...

The Gap safe screening technique is a powerful tool to accelerate the convergence of sparse optimization solvers. Its performance is largely based on the ability to determine the smallest “sphere”, centered at a given feasible dual point, that contains the dual solution. This can be achieved through an inner sphere refinement loop, applied at each...

Audio inpainting, i.e., the task of restoring missing or occluded audio signal samples, usually relies on sparse representations or autoregressive modeling. In this paper, we propose to structure the spectrogram with nonnegative matrix factorization (NMF) in a probabilistic framework. First, we treat the missing samples as latent variables, and der...

State-of-the-art music recommender systems are based on collaborative filtering, which builds upon learning similarities between users and songs from the available listening data. These approaches inherently face the cold-start problem, as they cannot recommend novel songs with no listening history. Content-aware recommendation addresses this issue...

Audio inpainting, i.e., the task of restoring missing or occluded audio signal samples, usually relies on sparse representations or autoregressive modeling. In this paper, we propose to structure the spectrogram with nonnegative matrix factorization (NMF) in a probabilistic framework. First, we treat the missing samples as latent variables, and der...

This paper tackles the problem of decomposing binary data using matrix factorization. We consider the family of mean-parametrized Bernoulli models, a class of generative models that are well suited for modeling binary data and enables interpretability of the factors. We factorize the Bernoulli parameter and consider an additional Beta prior on one...

Non-negative and bounded-variable linear regression problems arise in a variety of applications in machine learning and signal processing. In this paper, we propose a technique to accelerate existing solvers for these problems by identifying saturated coordinates in the course of iterations. This is akin to safe screening techniques previously prop...

This paper tackles the problem of decomposing binary data using matrix factorization. We consider the family of mean-parametrized Bernoulli models, a class of generative models that are well suited for modeling binary data and enables interpretability of the factors. We factorize the Bernoulli parameter and consider an additional Beta prior on one...

Non-negative matrix factorization with transform learning (TL-NMF) is a recent idea that aims at learning data representations suited to NMF. In this work, we relate TL-NMF to the classical matrix joint-diagonalization (JD) problem. We show that, when the number of data realizations is sufficiently large, TL-NMF can be replaced by a two-step approa...

This letter considers the phase retrieval (PR) problem, which aims to reconstruct a signal from phaseless measurements such as magnitude or power spectrograms. PR is generally handled as a minimization problem involving a quadratic loss. Recent works have considered alternative discrepancy measures, such as the Bregman divergences, but it is still...

Non-negative matrix factorization with transform learning (TL-NMF) is a recent idea that aims at learning data representations suited to NMF. In this work, we relate TL-NMF to the classical matrix joint-diagonalization (JD) problem. We show that, when the number of data realizations is sufficiently large, TL-NMF can be replaced by a two-step approa...

Many datasets are obtained as a resolution trade-off between two adversarial dimensions; for example between the frequency and the temporal resolutions for the spectrogram of an audio signal, and between the number of wavelengths and the spatial resolution for a hyper/multi-spectral image. To perform blind source separation using observations with...

This article proposes new multiplicative updates for nonnegative matrix factorization (NMF) with the $\beta$-divergence objective function. Our new updates are derived from a joint majorization-minimization (MM) scheme, in which an auxiliary function (a tight upper bound of the objective function) is built for the two factors jointly and minimized...

We consider an adversarially-trained version of the nonnegative matrix factorization, a popular latent dimensionality reduction technique. In our formulation, an attacker adds an arbitrary matrix of bounded norm to the given data matrix. We design efcient algorithms inspired by adversarial training to optimize for dictionary and coefcient matrices...

This paper addresses the problem of Unbalanced Optimal Transport (UOT) in which the marginal conditions are relaxed (using weighted penalties in lieu of equality) and no additional regularization is enforced on the OT plan. In this context, we show that the corresponding optimization problem can be reformulated as a non-negative penalized linear re...

We consider an adversarially-trained version of the nonnegative matrix factorization, a popular latent dimensionality reduction technique. In our formulation, an attacker adds an arbitrary matrix of bounded norm to the given data matrix. We design efficient algorithms inspired by adversarial training to optimize for dictionary and coefficient matri...

Positive semidefinite matrix factorization (PSDMF) expresses each entry of a nonnegative matrix as the inner product of two positive semidefinite (psd) matrices. When all these psd matrices are constrained to be diagonal, this model is equivalent to nonnegative matrix factorization. Applications include combinatorial optimization, quantum-based sta...

State-of-the-art music recommender systems are based on collaborative filtering, which builds upon learning similarities between users and songs from the available listening data. These approaches inherently face the cold-start problem, as they cannot recommend novel songs with no listening history. Content-aware recommendation addresses this issue...

Sparse optimization problems are ubiquitous in many fields such as statistics, signal/image processing and machine learning. This has led to the birth of many iterative algorithms to solve them. A powerful strategy to boost the performance of these algorithms is known as safe screening: it allows the early identification of zero coordinates in the...

Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of nonnegative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing wit...

Phase retrieval (PR) aims to recover a signal from the magnitudes of a set of inner products. This problem arises in many audio signal processing applications which operate on a short-time Fourier transform magnitude or power spectrogram, and discard the phase information. Recovering the missing phase from the resulting modified spectrogram is inde...

Phase retrieval aims to recover a signal from magnitude or power spectra measurements. It is often addressed by considering a minimization problem involving a quadratic cost function. We propose a different formulation based on Bregman divergences, which encompass divergences that are appropriate for audio signal processing applications. We derive...

Binary data matrices can represent many types of data such as social networks, votes, or gene expression. In some cases, the analysis of binary matrices can be tackled with nonnegative matrix factorization (NMF), where the observed data matrix is approximated by the product of two smaller nonnegative matrices. In this context, probabilistic NMF ass...

State-of-the-art music recommendation systems are based on collaborative filtering, which predicts a user's interest from his listening habits and similarities with other users' profiles. These approaches are agnostic to the song content, and therefore face the cold-start problem: they cannot recommend novel songs without listening history. To tack...

Time-frequency audio source separation is usually achieved by estimating the short-time Fourier transform (STFT) magnitude of each source, and then applying a phase recovery algorithm to retrieve time-domain signals. In particular, the multiple input spectrogram inversion (MISI) algorithm has shown good performance in several recent works. This alg...

Phase retrieval (PR) aims to recover a signal from the magnitudes of a set of inner products. This problem arises in many audio signal processing applications which operate on a short-time Fourier transform magnitude or power spectrogram, and discard the phase information. Recovering the missing phase from the resulting modified spectrogram is inde...

Non-negative matrix factorization with transform learning (TL-NMF) aims at estimating a short-time orthogonal transform that projects temporal data into a domain that is more amenable to NMF than off-the-shelf time-frequency transforms. In this work, we study the identifiability of TL-NMF under the Gaussian composite model. We prove that one can un...

Positive semidefinite matrix factorization (PSDMF) expresses each entry of a nonnegative matrix as the inner product of two positive semidefinite (psd) matrices. When all these psd matrices are constrained to be diagonal, this model is equivalent to nonnegative matrix factorization. Applications include combinatorial optimization, quantum-based sta...

Blind spectral unmixing is the problem of decomposing the spectrum of a mixed signal or image into a collection of source spectra and their corresponding activations indicating the proportion of each source present in the mixed spectrum. To perform this task, nonnegative matrix factorization (NMF) based on the $\beta$-divergence, referred to as $\b...

Non-negative matrix factorization (NMF) has become a well-established class of methods for the analysis of non-negative data. In particular, a lot of effort has been devoted to probabilistic NMF, namely estimation or inference tasks in probabilistic models describing the data, based for example on Poisson or exponential likelihoods. When dealing wi...

We introduce a new non-negative matrix factorization (NMF) method for ordinal data, called OrdNMF. Ordinal data are categorical data which exhibit a natural ordering between the categories. In particular, they can be found in recommender systems, either with explicit data (such as ratings) or implicit data (such as quantized play counts). OrdNMF is...

We propose a novel ranking model that combines the Bradley-Terry-Luce probability model with a nonnegative matrix factorization framework to model and uncover the presence of latent variables that influence the performance of top tennis players. We derive an efficient, provably convergent, and numerically stable majorization-minimization-based algo...

We introduce negative binomial matrix factorization (NBMF), a matrix factorization technique specially designed for analyzing over-dispersed count data. It can be viewed as an extension of Poisson factorization (PF) perturbed by a multiplicative term which models exposure. This term brings a degree of freedom for controlling the dispersion, making...

Count data are often used in recommender systems: they are widespread (song play counts, product purchases, clicks on web pages) and can reveal user preference without any explicit rating from the user. Such data are known to be sparse, over-dispersed and bursty, which makes their direct use in recommender systems challenging, often leading to pre-...

Factor analysis has proven to be a relevant tool for extracting tissue time-activity curves (TACs) in dynamic PET images, since it allows for an unsupervised analysis of the data. Reliable and interpretable results are possible only if considered with respect to suitable noise statistics. However, the noise in reconstructed dynamic PET images is ve...

We propose a novel ranking model that combines the Bradley-Terry-Luce probability model with a nonnegative matrix factorization framework to model and uncover the presence of latent variables that influence the performance of top tennis players. We derive an efficient, provably convergent, and numerically stable majorization-minimization-based algo...

Binary data matrices can represent many types of data such as social networks, votes or gene expression. In some cases, the analysis of binary matrices can be tackled with nonnegative matrix factorization (NMF), where the observed data matrix is approximated by the product of two smaller nonnegative matrices. In this context, probabilistic NMF assu...

Nonnegative matrix factorization (NMF) is a popular method for audio spectral unmixing. While NMF is traditionally applied to off-the-shelf time-frequency representations based on the short-time Fourier or Cosine transforms, the ability to learn transforms from raw data attracts increasing attention. However, this adds an important computational ov...

Temporal continuity is one of the most important features of time series data. In this chapter, we present several ways of modeling time dependencies in nonnegative matrix factorization (NMF). The dependencies between consecutive frames of the spectrogram can be imposed either on the basis matrix or on the activations. The former case is known as t...

Many state-of-the-art signal decomposition techniques rely on a low-rank factorization of a time-frequency (t-f) transform. In particular, nonnegative matrix factorization (NMF) of the spectrogram has been considered in many audio applications. This is an analysis approach in the sense that the factorization is applied to the squared magnitude of t...

Many state-of-the-art signal decomposition techniques rely on a low-rank factorization of a time-frequency (t-f) transform. In particular, nonnegative matrix factorization (NMF) of the spectrogram has been considered in many audio applications. This is an analysis approach in the sense that the factorization is applied to the squared magnitude of t...

This chapter introduces multichannel nonnegative matrix factorization (NMF) methods for audio source separation. All the methods and some of their extensions are introduced within a more general local Gaussian modeling (LGM) framework. These methods are very attractive since allow combining spatial and spectral cues in a joint and principal way, bu...

Spectral decomposition by nonnegative matrix factorisation (NMF) has become state-of-the-art practice in many audio signal processing tasks, such as source separation, enhancement or transcription. This chapter reviews the fundamentals of NMF-based audio decomposition, in unsupervised and informed settings. We formulate NMF as an optimisation probl...

We present novel understandings of the Gamma-Poisson (GaP) model, a probabilistic matrix factorization model for count data. We show that GaP can be rewritten free of the score/activation matrix. This gives us new insights about the estimation of the topic/dictionary matrix by maximum marginal likelihood estimation. In particular, this explains the...

Traditional NMF-based signal decomposition relies on the factorization of spectral data which is typically computed by means of the short-time Fourier transform. In this paper we propose to relax the choice of a pre-fixed transform and learn a short-time unitary transform together with the factorization, using a novel block-descent algorithm. This...

This work introduces a new framework for nonnegative matrix factorization (NMF) in multisensor or multimodal data configurations, where taking into account the mutual dependence that exists between the related parallel streams of data is expected to improve performance. In contrast with previous works that focused on co-factorization methods -where...

In this paper we propose a non-negative matrix factorization (NMF) model with piecewise-constant activation coefficients. This structure is enforced using a total variation penalty on the rows of the activation matrix. The resulting optimization problem is solved with a majorization-minimization procedure. The proposed algorithm is well suited to a...

This paper introduces a robust mixing model to describe hyperspectral data resulting from the mixture of several pure spectral signatures. This new model not only generalizes the commonly used linear mixing model, but also allows for possible nonlinear effects to be easily handled, relying on mild assumptions regarding these nonlinearities. The sta...

Non-negative data arise in a variety of important signal processing domains, such as power spectra of signals, pixels in images, and count data. This paper introduces a novel non-negative dynamical system (NDS) for sequences of such data, and describes its application to modeling speech and audio power spectra. The NDS model can be interpreted both...

This paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization (NMF) with the $(\beta)$--divergence. The $(\beta)$-divergence is a family of cost functions that includes the squared euclidean distance, Kullback-Leibler (KL) and Itakura-Saito (IS) divergences as special cases. Learning the model order is importa...

This paper introduces a robust linear model to describe hyperspec-tral data arising from the mixture of several pure spectral signatures. This new model not only generalizes the commonly used linear mixing model but also allows for possible nonlinear effects to be handled, relying on mild assumptions regarding these nonlin-earities. Based on this m...

This paper presents a new method for bimodal nonnegative matrix factorization (NMF). This method is well-suited to situations where two streams of data are concurrently analyzed and are expected to be related by loosely common factors. It allows for a soft co-factorization, which takes into account the relationship that exists between the modalitie...

We introduce a novel video structuring scheme that exploits nonnegative matrix factorization (NMF) on count data (in a bag of features representation of the visual stream) to jointly discover latent structuring patterns and their activations in time. Our NMF variant employs the Kullback-Leibler divergence as a cost function and imposes a temporal s...

There has been a significant amount of research in new algorithms and applications for nonnegative matrix factorization, but relatively little has been published on practical considerations for real-world applications, such as choosing optimal parameters for a particular application. In this paper, we will look at two applications, single-channel s...

This paper introduces a new paradigm for unsupervised audiovisual document structuring. In this paradigm, a novel Nonnegative Matrix Factorization (NMF) algorithm is applied on histograms of counts (relating to a bag of features representation of the content) to jointly discover latent structuring patterns and their activations in time. Our NMF var...

This paper describes a probabilistic approach to template-based chord recognition in music signals. The algorithm only takes chromagram data and a user-defined dictionary of chord templates as input data. No training or musical information such as key, rhythm, or chord transition models is required. The chord occurrences are treated as probabilisti...

Nonnegative matrix factorization (NMF) with the Itakura-Saito divergence has proven efficient for audio source separation and music transcription, where the signal power spectrogram is factored into a "dictionary" matrix times an "activation" matrix. Given the nature of audio signals it is expected that the activation coefficients exhibit smoothnes...

We describe an alternative to standard nonnegative matrix factorisation (NMF) for nonnegative dictionary learning. NMF with the Kullback-Leibler divergence can be seen as maximisation of the joint likelihood of the dictionary and the expansion coefficients under Poisson observation noise. This approach lacks optimality be cause the number of parame...

Nonnegative matrix factorization (NMF) is now a common tool for audio source separation. When learning NMF on large audio databases, one major drawback is that the complexity in time is O(FKN) when updating the dictionary (where (F;N) is the dimension of the input power spectrograms, and K the number of basis spectra), thus forbidding its applicati...

This letter describes algorithms for nonnegative matrix factorization (NMF) with the β-divergence (β-NMF). The β-divergence is a family of cost functions parameterized by a single shape parameter β that takes the Euclidean distance, the Kullback-Leibler divergence, and the Itakura-Saito divergence as special cases (β = 2, 1, 0 respectively). The pr...

Space alternating data augmentation (SADA) was proposed by Doucet et al (2005) as a MCMC generalization of the SAGE algorithm of Fessler and Hero (1994), itself a famous variant of the EM algorithm. While SADA had previously been applied to inference in Gaussian mixture models, we show this sampler to be particularly well suited for models having a...

We propose an unsupervised inference procedure for audio source separation. Components in nonnegative matrix factorization (NMF) are grouped automatically in audio sources via a penalized maximum likelihood approach. The penalty term we introduce favors sparsity at the group level, and is motivated by the assumption that the local amplitude of the...

Separating multiple tracks from professionally produced music recordings (PPMRs) is still a challenging problem. We address this task with a user-guided approach in which the separation system is provided segmental information indicating the time activations of the particular instruments to separate. This information may typically be retrieved from...

This paper describes a method for chord recognition from audio signals. Our method provides a coherent and relevant probabilistic framework for template-based transcription. The only information needed for the transcription is the definition of the chords : in particular neither annotated audio data nor music theory knowledge is required. We extrac...

Nonnegative tensor factorization (NTF) of multichannel spectrograms under PARAFAC structure has recently been proposed by Fitzgerald et al as a mean of performing blind source separation (BSS) of multichannel audio data. In this paper we investigate the statistical source models implied by this approach. We show that it implicitly assumes a nonpoin...

This paper describes algorithms for nonnegative matrix factorization (NMF) with the beta-divergence (beta-NMF). The beta-divergence is a family of cost functions parametrized by a single shape parameter beta that takes the Euclidean distance, the Kullback-Leibler divergence and the Itakura-Saito divergence as special cases (beta = 2,1,0, respective...

We consider inference in a general data-driven object-based model of multichannel audio data, assumed generated as a possibly underdetermined convolutive mixture of source signals. We work in the short-time Fourier transform (STFT) domain, where convolution is routinely approximated as linear instantaneous mixing in each frequency band. Each source...

Extracting the main melody from a polyphonic music recording seems natural even to untrained human listeners. To a certain extent it is related to the concept of source separation, with the human ability of focusing on a specific source in order to extract relevant information. In this paper, we propose a new approach for the estimation and extract...

Nonnegative matrix factorization (NMF) is a popular linear regression technique in the fields of machine learning and signal/image processing. Much research about this topic has been driven by applications in audio. NMF has been for example applied with success to automatic music transcription and audio source separation, where the data is usually...

We present a new probabilistic model for polyphonic audio termed factorial scaled hidden Markov model (FS-HMM), which generalizes several existing models, notably the Gaussian scaled mixture model and the Itakura-Saito nonnegative matrix factorization (NMF) model. We describe two expectation-maximization (EM) algorithms for maximum likelihood estim...

This paper presents an efficient method for chord transcription of music signals. A succession of chroma vectors is calculated from the signal in order to extract the musical content of the piece over time. We introduce a set of chord templates for several types of chords (major, minor, dominant seventh, ...): different chord models taking into acc...

Nonnegative matrix factorization (NMF) has become a popular technique for data analysis and dimensionality reduction. However, it is often assumed that the number of latent dimensions (or components) is given. In practice, one must choose a suitable value depending on the data and/or setting. In this paper, we address this important issue by using...

In this paper we are interested in non-negative matrix factorization (NMF) with the Itakura-Saito (IS) divergence. Previous work has demonstrated the relevance of this cost function for the decompo- sition of audio power spectrograms. This is in particular due to its scale invariance, which makes it more robust to the wide dynamics of audio, a prop...

We consider inference in a general data-driven object-based model of multichannel audio data, assumed generated as a possibly under- determined convolutive mixture of source signals. Each source is given a model inspired from nonnegative matrix factorization (NMF) with the Itakura-Saito divergence, which underlies a statistical model of superimpose...

This paper describes a fast and efficient template-based chord recognition method. We introduce three chord mod- els taking into account one or more harmonics for the notes of the chord. The use of pre-determined chord models enables to consider several types of chords (major, mi- nor, dominant seventh, minor seventh, augmented, dimin- ished...). A...

This paper describes a fast and efficient template-based chord recognition method. We introduce three chord mod-els taking into account one or more harmonics for the notes of the chord. The use of pre-determined chord models en-ables to consider several types of chords. After extracting a chromagram from the signal, the detected chord over a frame...

We develop an interpretation of nonnegative matrix factorization (NMF) methods based on Euclidean distance, Kullback-Leibler and Itakura-Saito divergences in a probabilistic framework. We describe how these factorizations are implicit in a well-defined statistical model of superimposed components, either Gaussian or Poisson distributed, and are equ...

We propose a new approach to solo/accompaniment separation from stereophonic music recordings which extends a monophonic algo- rithm we recently proposed. The solo part is modelled using a source/filter model to which we added two contributions: an ex- plicit smoothing strategy for the filter frequency responses and an unvoicing model to catch the...

This letter presents theoretical, algorithmic, and experimental results about nonnegative matrix factorization (NMF) with the Itakura-Saito (IS) divergence. We describe how IS-NMF is underlaid by a well-defined statistical model of superimposed gaussian components and is equivalent to maximum likelihood estimation of variance parameters. This setti...