ℓ1\ell^1-Analysis Minimization and Generalized (Co-)Sparsity: When Does Recovery Succeed?

This paper investigates the problem of signal estimation from undersampled noisy sub-Gaussian measurements under the assumption of a cosparse model. Based on generalized notions of sparsity, we derive novel recovery guarantees for the $\ell^{1}$-analysis basis pursuit, enabling highly accurate predictions of its sample complexity. The corresponding bounds on the number of required measurements do explicitly depend on the Gram matrix of the analysis operator and therefore particularly account for its mutual coherence structure. Our findings defy conventional wisdom which promotes the sparsity of analysis coefficients as the crucial quantity to study. In fact, this common paradigm breaks down completely in many situations of practical interest, for instance, when applying a redundant (multilevel) frame as analysis prior. By extensive numerical experiments, we demonstrate that, in contrast, our theoretical sampling-rate bounds reliably capture the recovery capability of various examples, such as redundant Haar wavelets systems, total variation, or random frames. The proofs of our main results build upon recent achievements in the convex geometry of data mining problems. More precisely, we establish a sophisticated upper bound on the conic Gaussian mean width that is associated with the underlying $\ell^{1}$-analysis polytope. Due to a novel localization argument, it turns out that the presented framework naturally extends to stable recovery, allowing us to incorporate compressible coefficient sequences as well

Hearing in a shoe-box : binaural source position and wall absorption estimation using virtually supervised learning

International audienceThis paper introduces a new framework for supervised sound source localization referred to as virtually-supervised learning. An acoustic shoe-box room simulator is used to generate a large number of binaural single-source audio scenes. These scenes are used to build a dataset of spatial binaural features annotated with acoustic properties such as the 3D source position and the walls' absorption coefficients. A probabilistic high- to low-dimensional regression framework is used to learn a mapping from these features to the acoustic properties. Results indicate that this mapping successfully estimates the azimuth and elevation of new sources, but also their range and even the walls' absorption coefficients solely based on binaural signals. Results also reveal that incorporating random-diffusion effects in the data significantly improves the estimation of all parameters

Physics-driven inverse problems made tractable with cosparse regularization

International audienceSparse data models are powerful tools for solving ill-posed inverse problems. We present a regularization framework based on the sparse synthesis and sparse analysis models for problems governed by linear partial differential equations. Although nominally equivalent, we show that the two models differ substantially from a computational perspective: unlike the sparse synthesis model, its analysis counterpart has much better scaling capabilities and can indeed be faster when more measurement data is available. Our findings are illustrated on two examples, sound source localization and brain source localization, which also serve as showcases for the regularization framework. To address this type of inverse problems, we develop a specially tailored convex optimization algorithm based on the Alternating Direction Method of Multipliers

PHYSALIS (Physics-Driven Cosparse Analysis)

PHYSALIS (Physics-Driven Cosparse Analysis) is a software package for reproducible research. It contains the Matlab routines allowing to reproduce the main experimental results on joint source localization and estimation of the Ph.D. thesis of Srdan Kitic, notably of the articleSrđan Kitić, Laurent Albera, Nancy Bertin, Rémi Gribonval. Physics-driven inverse problems made tractable with cosparse regularization. IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 2016, 64 (2), pp.335-34

PHYSALIS (Physics-Driven Cosparse Analysis)

PHYSALIS (Physics-Driven Cosparse Analysis) is a software package for reproducible research. It contains the Matlab routines allowing to reproduce the main experimental results on joint source localization and estimation of the Ph.D. thesis of Srdan Kitic, notably of the articleSrđan Kitić, Laurent Albera, Nancy Bertin, Rémi Gribonval. Physics-driven inverse problems made tractable with cosparse regularization. IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 2016, 64 (2), pp.335-34