Sparsity and adaptivity for the blind separation of partially correlated sources

Blind source separation (BSS) is a very popular technique to analyze multichannel data. In this context, the data are modeled as the linear combination of sources to be retrieved. For that purpose, standard BSS methods all rely on some discrimination principle, whether it is statistical independence or morphological diversity, to distinguish between the sources. However, dealing with real-world data reveals that such assumptions are rarely valid in practice: the signals of interest are more likely partially correlated, which generally hampers the performances of standard BSS methods. In this article, we introduce a novel sparsity-enforcing BSS method coined Adaptive Morphological Component Analysis (AMCA), which is designed to retrieve sparse and partially correlated sources. More precisely, it makes profit of an adaptive re-weighting scheme to favor/penalize samples based on their level of correlation. Extensive numerical experiments have been carried out which show that the proposed method is robust to the partial correlation of sources while standard BSS techniques fail. The AMCA algorithm is evaluated in the field of astrophysics for the separation of physical components from microwave data.

NMF with Sparse Regularizations in Transformed Domains

Non-negative blind source separation (non-negative BSS), which is also referred to as non-negative matrix factorization (NMF), is a very active field in domains as different as astrophysics, audio processing or biomedical signal processing. In this context, the efficient retrieval of the sources requires the use of signal priors such as sparsity. If NMF has now been well studied with sparse constraints in the direct domain, only very few algorithms can encompass non-negativity together with sparsity in a transformed domain since simultaneously dealing with two priors in two different domains is challenging. In this article, we show how a sparse NMF algorithm coined non-negative generalized morphological component analysis (nGMCA) can be extended to impose non-negativity in the direct domain along with sparsity in a transformed domain, with both analysis and synthesis formulations. To our knowledge, this work presents the first comparison of analysis and synthesis priors ---as well as their reweighted versions--- in the context of blind source separation. Comparisons with state-of-the-art NMF algorithms on realistic data show the efficiency as well as the robustness of the proposed algorithms.

Sparse and Non-Negative BSS for Noisy Data

Non-negative blind source separation (BSS) has raised interest in various fields of research, as testified by the wide literature on the topic of non-negative matrix factorization (NMF). In this context, it is fundamental that the sources to be estimated present some diversity in order to be efficiently retrieved. Sparsity is known to enhance such contrast between the sources while producing very robust approaches, especially to noise. In this paper we introduce a new algorithm in order to tackle the blind separation of non-negative sparse sources from noisy measurements. We first show that sparsity and non-negativity constraints have to be carefully applied on the sought-after solution. In fact, improperly constrained solutions are unlikely to be stable and are therefore sub-optimal. The proposed algorithm, named nGMCA (non-negative Generalized Morphological Component Analysis), makes use of proximal calculus techniques to provide properly constrained solutions. The performance of nGMCA compared to other state-of-the-art algorithms is demonstrated by numerical experiments encompassing a wide variety of settings, with negligible parameter tuning. In particular, nGMCA is shown to provide robustness to noise and performs well on synthetic mixtures of real NMR spectra.

Multivariate Temporal Dictionary Learning for EEG

This article addresses the issue of representing electroencephalographic (EEG) signals in an efficient way. While classical approaches use a fixed Gabor dictionary to analyze EEG signals, this article proposes a data-driven method to obtain an adapted dictionary. To reach an efficient dictionary learning, appropriate spatial and temporal modeling is required. Inter-channels links are taken into account in the spatial multivariate model, and shift-invariance is used for the temporal model. Multivariate learned kernels are informative (a few atoms code plentiful energy) and interpretable (the atoms can have a physiological meaning). Using real EEG data, the proposed method is shown to outperform the classical multichannel matching pursuit used with a Gabor dictionary, as measured by the representative power of the learned dictionary and its spatial flexibility. Moreover, dictionary learning can capture interpretable patterns: this ability is illustrated on real data, learning a P300 evoked potential.