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              <p>Sparse regressions to solve ill-posed inverse problems have been massively investigated over the last decade. Yet, when noise is present in the model, it is almost exclusively considered as Gaussian and white. While this assumption can hold in practice it rarely holds when observations are time series as they are corrupted by auto-correlated and colored noise. In this work we study sparse regression under the assumption of non white Gaussian noise and explain how to run the inference using proximal gradient methods. We investigate an application in brain imaging: the problem of source localization using magneto- and electroencephalography (M/EEG) which allow functional brain imaging with high temporal resolution. We use a time-frequency representation of the source waveforms and a sparse regularization which promotes focal sources with smooth and transient activations. Our approach is evaluated using simulations comparing it to strategies that assume the noise is white or to simple prewhitening.</p>
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              <p>Les régularisations parcimonieuses pour résoudre des problèmes inverses mal-posés ont été massivement étudiées ces dix dernières années. Jusqu'à présent, lorsque le bruit est pris en compte dans le modèle, il est presque systématiquement considéré comme blanc Gaussien. Bien que cette hypothèse peut être vérifiée en pratique, elle ne l'est que rarement lorsque les observations sont des séries temporelles dégradées par des bruits autocorrélés. Dans cet article, on étudie la régression parcimonieuse sous une hypothèse de bruit Gaussien non blanc, et comment inférer les paramètres du modèle avec les méthodes proximales. On étudie une application d'imagerie cérébrale: le problème de localisation de sources basée sur la magnéto-et l'électroencephalographie (M/EEG) ce qui permet l'imagerie fonctionnelle du cerveau avec une résolution temporelle élevée. On utilise pour cela une représentation temps-fréquence des sources et une régularisation parcimonieuse qui favorise les sources focales avec des activations régulières et transitoires. Notre approche est évaluée sur des simulations en comparant plusieurs stratégies qui suppose que le bruit est blanc, ou qui utilise un simple pré-blanchiment. Abstract – Sparse regressions to solve ill-posed inverse problems have been massively investigated over the last decade. Yet, when noise is present in the model, it is almost exclusively considered as Gaussian and white. While this assumption can hold in practice it rarely holds when observations are time series as they are corrupted by auto-correlated and colored noise. In this work we study sparse regression under the assumption of non white Gaussian noise and explain how to run the inference using proximal gradient methods. We investigate an application in brain imaging: the problem of source localization using magneto-and electroencephalography (M/EEG) which allow functional brain imaging with high temporal resolution. We use a time-frequency representation of the source waveforms and a sparse regularization which promotes focal sources with smooth and transient activations. Our approach is evaluated using simulations comparing it to strategies that assume the noise is white or to simple prewhitening.</p>
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