Section: New Results

Emerging activities on compressive learning and inverse problems

Compressive sensing, compressive learning, audio inpainting,

Audio inpainting

Participants : Rémi Gribonval, Nancy Bertin, Srdan Kitic.

Inpainting is a particular kind of inverse problems that has been extensively addressed in the recent years in the field of image processing.

Building upon our previous pioneering 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 [59] ), new results were obtained last year and this year to address the case of audio declipping with the competitive cosparse approach. Last year, 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 [95] , which is a counterpart of the sparse Consistent Iterative Hard Thresholding.

This year, we proposed a new algorithmic framework called SPADE, based on non-convex heuristics and which can accomodate both the sparse and cosparse prior. We studied their performance numerically and observed in particular that its cosparse version offers a very appealing trade-off between reconstruction performance and computational time [31] , making it suitable for practical applications, even in real-time. We could also confirm our results by subjective listening tests conducted this year [12] .

The work on cosparse audio declipping was awarded the Conexant best paper award at the LVA/ICA conference [31] and draw the attention of a world leading company in professional audio signal processing, with which some transfer has been negotiated.

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, joint source separation and declipping).

Blind Calibration of Impedance and Geometry

Participants : Rémi Gribonval, Nancy Bertin, Srdan Kitic.

Main collaborations: Laurent Daudet, Thibault Nowakowski, Julien de Rosny (Institut Langevin)

This year, we also investigated extended inverse problem scenarios where a “lack of calibration” may occur, i.e., when some physical parameters are needed for reconstruction but apriori unknown: speed of sound, impedance at the boundaries of the domain where the studied phenomenon propagates, or even the shape of these boundaries. In a first approach, based on our physics-driven cosparse regularization of the sound source localization problem [23] (see section  7.1.3 ), we managed to preserve the sound source localization performance when the speed of sound is unkonwn, or, equally, when the impedance is unknown, provided the shape is and under some smoothness assumptions. Unlike the previous case (gain calibration), the arising problems are not convex but biconvex, and can be solved with proper biconvex formulation of ADMM algorithm. In a second approach based on eigenmode decomposition (limited to a 2D membrane), we showed that impedance learning with known shape, or shape learning with known impedance can be expressed as two facets of the same problem, and solved by the same approach, from a small number of measurements. Two papers presenting these two sets of results were accepted for publication in ICASSP 2016 [38] , [35] .

Sketching for Large-Scale Mixture Estimation

Participants : Rémi Gribonval, Nicolas Keriven.

Main collaborations: Patrick Perez (Technicolor R&I France) Anthony Bourrier (formerly Technicolor R&I France, now at GIPSA-Lab)

When fitting a probability model to voluminous data, memory and computational time can become prohibitive. In this work, 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 [70] , [68] , [69] . This year, we consolidated our extensions to non-isotropic Gaussians, with new algorithms [49] and conducted large-scale experiments demonstrating its potential for speaker verification. A conference paper has been accepted to ICASSP 2016 [40] and a journal version is being finalized.