Section: New Results

Emerging activities on compressive sensing, learning and inverse problems

Compressive sensing, compressive learning, acoustic wavefields, audio inpainting,

Audio inpainting

Participants : Rémi Gribonval, Nancy Bertin, Corentin Guichaoua, Srdan Kitic, Anh Tho Le.

Inpainting is a particular kind of inverse problems that has been extensively addressed in the recent years in the field of image processing. It consists in reconstructing a set of missing pixels in an image based on the observation of the remaining pixels. Sparse representations have proved to be particularly appropriate to address this problem. However, inpainting audio data has never been defined as such so far. A series of works about audio inpainting was initiated by the METISS team in the framework of the EU Framework 7 FET-Open project FP7-ICT-225913-SMALL (Sparse Models, Algorithms and Learning for Large-Scale data).

Building upon our previous contributions (definition of the audio inpainting problem as a general framework for many audio processing tasks, application to the audio declipping or desaturation problem, formulation as a sparse recovery problem [60] ), new results were obtained this year to address the case of audio declipping with the competitive cosparse approach. Its promising results, especially when the clipping level is low, were confirmed experimentally by the formulation and use of a new algorithm named Cosparse Iterative Hard Tresholding, which is a counterpart of the sparse Consistent Iterative Hard Thresholding. These results were presented during the iTwist'14 workshop [49] . Additional experiments were performed (internship of Anh Tho Le) to confirm the results on a larger database and investigate optimal parameters (nature and redundancy of the dictionary, relaxation parameter for the cosparsity level).

Current and future works deal with developing advanced (co)sparse decomposition for audio inpainting, including several forms of structured sparsity (e.g. temporal and multichannel joint-sparsity), dictionary learning for inpainting, and several applicative scenarios (declipping, time-frequency inpainting).

Blind Calibration of Compressive Sensing systems

Participants : Rémi Gribonval, Cagdas Bilen, Gilles Puy.

Main collaborations: Gilles Chardon, Laurent Daudet (Institut Langevin)

We consider the problem of calibrating a compressed sensing measurement system under the assumption that the decalibration consists of unknown gains on each measure. We focus on blind calibration, using measures performed on a few unknown (but sparse) signals. A naive formulation of this blind calibration problem, using ${\ell }^1$ minimization, is reminiscent of blind source separation and dictionary learning, which are known to be highly non-convex and riddled with local minima. In the considered context, when the gains are real valued and non-negative, we showed that in fact this formulation can be exactly expressed as a convex optimization problem, and can be solved using off-the-shelf algorithms. Numerical simulations demonstrated the effectiveness of the approach even for highly uncalibrated measures, when a sufficient number of (unknown, but sparse) calibrating signals is provided. We observed that the success/failure of the approach seems to obey sharp phase transitions  [84] . We extended the framework to phase-only decalibration, using techniques revolving around low-rank matrix recovery [66] , [65] , [110] , [64] , and to joint phase and gain decalibration [15] .

Compressive Gaussian Mixture estimation

Participants : Rémi Gribonval, Anthony Bourrier, Nicolas Keriven.

Main collaborations: Patrick Perez (Technicolor R&I France)

When fitting a probability model to voluminous data, memory and computational time can become prohibitive. In this paper, we propose a framework aimed at fitting a mixture of isotropic Gaussians to data vectors by computing a low-dimensional sketch of the data. The sketch represents empirical moments of the underlying probability distribution. Deriving a reconstruction algorithm by analogy with compressive sensing, we experimentally show that it is possible to precisely estimate the mixture parameters provided that the sketch is large enough. Our algorithm provides good reconstruction and scales to higher dimensions than previous probability mixture estimation algorithms, while consuming less memory in the case of numerous data. It also provides a privacy-preserving data analysis tool, since the sketch does not disclose information about individual datum it is based on [71] , [69] , [70] . This year, extensions to non-isotropic Gaussians, with new algorithms and preliminary applications to speaker verification have been conducted.