Section: New Results
Emerging activities on Compressive Learning and Nonlinear Inverse Problems
Compressive sensing, compressive learning, audio inpainting, phase estimation
Phase Estimation in Multichannel Mixtures
Participants : Antoine Deleforge, Yann Traonmilin.
The problem of estimating source signals given an observed multichannel mixture is fundamentally ill-posed when the mixing matrix is unknown or when the number of sources is larger that the number of microphones. Hence, prior information on the desired source signals must be incorporated in order to tackle it. An important line of research in audio source separation over the past decade consists in using a model of the source signals' magnitudes in the short-time Fourier domain . Such models can be inferred through, e.g., non-negative matrix factorization  or deep neural networks . Magnitudes estimates are often interpreted as instantaneous variances of Gaussian-process source signals, and are combined with Wiener filtering for source separation. In , we introduced a shift of this paradigm by considering the Phase Unmixing problem: how can one recover the instantaneous phases of complex mixed source signals when their magnitudes and mixing matrix are known? This problem was showed to be NP-hard, and three approaches were proposed to tackle it: a heuristic method, an alternate minimization method, and a convex relaxation into a semi-definite program. The last two approaches were showed to outperform the oracle multichannel Wiener filter in under-determined informed source separation tasks. The latter yielded best results, including the potential for exact source separation in under-determined settings.
Audio Inpainting and Denoising
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 ), we proposed over the last two years a series of algorithms leveraging the competitive cosparse approach, which offers a very appealing trade-off between reconstruction performance and computational time , , . The work on cosparse audio declipping which was awarded the Conexant best paper award at the LVA/ICA 2015 conference , together with the associated toolbox for reproducible research (see Section 6.8) draw the attention of a world leading company in professional audio signal processing, with which some transfer has been negotiated. In 2016, real-time implementation of the A-SPADE algorithm was obtained and demonstrated at various events (HCERES evaluation, TechnofÃ©rence # 18 « Nouvelles expériences son et vidéo », ...).
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, denoising, time-frequency inpainting, joint source separation and declipping). In particular, we investigated the incorporation of the so-called “social” structure constraint  into problems regularized by a cosparse prior, including declipping and denoising. Publication of this work is currently under preparation.
Blind Calibration of Impedance and Geometry
Participants : Rémi Gribonval, Nancy Bertin, Srdan Kitic.
Main collaborations: Laurent Daudet, Thibault Nowakowski, Julien de Rosny (Institut Langevin)
Last 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  (see section 7.1.2), we managed to preserve the sound source localization performance when the speed of sound is unknown, 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 appeared at ICASSP 2016 , .
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, then GIPSA-Lab)
When fitting a probability model to voluminous data, memory and computational time can become prohibitive. We proposed during the Ph.D. thesis of Anthony Bourrier  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 showed that it is possible to precisely estimate the mixture parameters provided that the sketch is large enough. The proposed algorithm provided good reconstruction and scaled to higher dimensions than previous probability mixture estimation algorithms, while consuming less memory in the case of voluminous datasets. It also provided a potentially privacy-preserving data analysis tool, since the sketch does not explicitly disclose information about individual datum it is based on , , . Last year, we consolidated our extensions to non-isotropic Gaussians, with new algorithms  and conducted large-scale experiments demonstrating its potential for speaker verification. A conference paper appeared at ICASSP 2016  and a journal version has been submitted , accompanied by a toolbox for reproducible research (see Section 6.12).
This year the work concentrated on extending the approach beyond the case of Gaussian Mixture Estimation. First, we showed empirically that the algorithm can be adapted to sketch a training collection while still allowing to compute clusters. The approach, called “Compressive K-means”, is described in a paper accepted at ICASSP 2017 . Then, we expressed a theoretical framework for sketched learning, encompassing statistical learning guarantees as well as dimension reduction guarantees. The framework already covers compressive K-means as well as compressive Principal Component Analysis (PCA), and a conference paper has been submitted. A comprehensive journal paper is under preparation, and future work will include expliciting the impact of the proposed framework on a wider set of concrete learning problems.