•  Signal and Image Processing
•  Machine Learning
•  Low-rank Matrix Approximations
•  Nonnegative Matrix Factorization (NMF)
•  Sparse Component Analysis (SCA)
•  Sparsity
•  Missing values
•  Time Delay of Arrival (TDOA)
•  Compressive Learning
•  Big Data
•  Blind and Informed Source Separation
•  Multiple Source Counting and Localization
•  In Situ Sensor Calibration
•  Applications: audio signals, hyperspectral data, chemics, ...

A matrix is low-rank if the number of latent variables that may explain it is (much) lower than the dimension of the matrix. While classical low-rank approximations rely on the Singular Value Decomposition (SVD) or the Principal Component Analysis (PCA), matrix factorization is another popular technique which is based on low-rankness. In particular, Nonnegative Matrix Factorization (NMF) is a very popular machine learning tool which finds many applications in signal and image processing. It aims to factor a given nonnegative data matrix as the product of two nonnegative matrix factors. It also allows, in its weighted form, to perform the completion of the missing entries in the data matrix.

Even if it provides very interesting and interpretable results, in its standard version, NMF is not guaranteed to converge to a global solution. This is particularly important in some problems for which the NMF outputs are very sensitive to the initialization or when the matrix factors are structured by the problem formalism. We thus proposed several informed NMF methods where additional knowledge is taken into account as a specific parameterization and/or as penalization terms in the NMF cost function.

Such proposed techniques were sucessfully applied to industrial source apportionment. We introduced several approaches which add extra-information (provided by experts in the framework of the considered application) as constraints in the NMF procedure. We also used a physical pollution propagation model to inform the NMF procedure and discard the unactive sources. Our informed NMF methods were shown to outperform the state-of-the-art NMF approaches. This work was realized in the framework of the Ph.D. theses of Abdelhakim LIMEM and Robert CHREIKY, respectively, in collaboration with Gilles DELMAIRE, Gilles ROUSSEL, Mahmoud OMIDVAR, Frédéric LEDOUX, and Dominique COURCOT.

We also extended such techniques for in situ sensor calibration which aims to compensate sensor drift through data-driven approaches. For that purpose, we revisited in situ calibration of a mobile sensor network as an informed matrix factorization problem with missing data. The main advantage of our formalism lies in the fact that we highly relax the mobile sensor rendezvous assumption with respect to state-of-the-art techniques. In the framework of the Ph.D. theses of Clément DORFFER and Farouk YAHAYA, we aim to apply such techniques to mobile crowdsensing, i.e., when a lot of mobile sensors (provided by mobile devices such as smartphones) are aquiring some information and are sharing it. In particular, we investigate such approaches for air quality monitoring:

• in the framework of the OSCAR project, funded by the "Chercheurs Citoyens" program of the Région Hauts-de-France and part of the REPERE program of the French Ecology Ministry, in which we were involved together with ATMO, BES, and Romain ROUVOY's group at Spirals Inria team.
• in the framework of a CNRS MASTODONS project in which we were involved with Spirals Inria team and the PC2A laboratory.

Random projections (a.k.a matrix sketching a.k.a compressive learning) belong to the major techniques used to process big data. Assuming the data to process to be low-rank, they allow to reduce their size with a negligible loss of information, and thus to speed-up the computations. They have been successfully applied to, e.g., NMF. In the framework of the Ph.D. thesis of Farouk YAHAYA, we are interested in their applications to the situation when some entries of a data matrix are missing (and more generally to the case when some confidence measures are associated with each data entry). We proposed specific compressed weighted NMF methods and found them to significanlty speed-up state-of-the-art methods, under some mild conditions.

We recently considered demosaicing for snapshot spectral imaging, in the framework of the Ph.D. thesis of Kinan ABBAS. While this problem was revisited as a low-rank completion method, we proposed to jointly demosaice and unmix such images. In particular, in addition to a naive WNMF approach, we proposed techniques which combine the ideas of WNMF with Sparse Component Analysis (SCA). In particular, we assume the existance of rank-one areas where only one endmember is present. Our proposed methods outperform the unmixing performance of two-stage approaches (consisting of running a state-of-the-art demosaicing method and then unmixing the recovered multispectral datacube) while providing a similar demosaicing performance.

Blind Source Separation (BSS) consists of estimating a set of unknown sources from a set of observations resulting from mixtures of these sources through unknown propagation channels. While most of the methods of the literature suppose the sources to be statistically independent, some approaches assume that sources are sparse in a chosen analysis domain, hence their name of Sparse Component Analysis (SCA) methods. SCA approaches generally suppose that sources have disjoint supports in the analysis domain, which is a strong sparsity assumption. On the contrary, Yannick DEVILLE's team (in which I was preparing my Ph.D. thesis) proposed methods which highly relax these assumptions. We improved and studied such methods for linear instantaneous mixtures, which is the simplest class of mixtures. We then extended these methods to more general mixture models, while still keeping low sparsity assumptions, and later to images. In particular, we studied the contribution provided by clustering techniques to our extensions. We mainly applied these approaches to audio signals (speech and music) and astrophysical data (interstellar dust spectra) and we showed that the performance achieved by them outperformed the one of classical BSS approaches.

In the framework of the AVID-MODE IAPP Marie-Curie FP7 project, under the supervision of Athanasios MOUCHTARIS, I was involved in the development of audio BSS methods for mobile devices. In particular, these devices contain small microphones which provide nonlinearities such as saturation: source signals are first linearly mixed, providing linear observations for each of which a component-wise nonlinear function, modeling the microphone nonlinearity, is applied. The resulting mixture model is known under the name of "post-nonlinear". I specifically proposed extensions of the above linear sparse component analysis methods to nonlinear mixtures.

I have also been involved in a project, in collaboration with Despoina PAVLIDI, Anthony GRIFFIN, and Athanasios MOUCHTARIS about real-time adaptative multiple source localization and counting using a circular microphone array. The proposed approach combined tools inheritated from SCA, with single-source Direction-of-Arrival (DOA) estimation methods, and Matching Pursuit, and was shown to be effective in various scenarios (see youtube demo).

I also worked, in collaboration with Emmanuel VINCENT and Yannick DEVILLE, on the validity of the independence assumption (stated and needed in Independent Component Analysis), when applied to speech and music signals. In particular, We investigated some dependency measures, in the temporal and the time-frequency domains, resp. in the framework of instantaneous and convolutive mixtures. Moreover, we tested several ICA methods, based on the above dependency measures, on the same source signals. We experimentally showed that speech and music sources tend to have the same mean behaviour for excerpt durations above 20 ms, but music signals provide more spread dependency measures and SIR values. Lastly, we experimentally showed that Gaussian nonstationary mutual information is better suited to audio signals than mutual information.

Later, Emmanuel VINCENT, Yannick DEVILLE, Anthony GRIFFIN, Athanasios MOUCHTARIS, and I investigated the influence of lossy audio encoding methods on the ICA performance. We showed that, even if the bit rate is high, the spatial diversity due to MP3 stereo coding has serious effects on the performance of classical ICA methods while the spectro-temporal filtering due to mono compression has almost no effect. We finally concluded to the need of new generation BSS methods which take into account the audio encoding procedure, e.g., for immersive audio systems (providing tens of observations).

• LI-TIFROM: TIme-Frequency Ratio of Mixtures BSS method for Linear-Instantaneous mixtures
• Faster-than-fast NMF: Nonnegative Matrix Factorization combining random projections and Nesterov iterations
• (AC/Sp/SpA)IN-Cal: Informed Nonnegative Matrix Factorization Methods for Mobile Sensor Calibration