Sparse BSS in the presence of outliers

While real-world data are often grossly corrupted, most techniques of blind source separation (BSS) give erroneous results in the presence of outliers. We propose a robust algorithm that jointly estimates the sparse sources and outliers without requiring any prior knowledge on the outliers. More precisely, it uses an alternative weighted scheme to weaken the influence of the estimated outliers. A preliminary experiment is presented and demonstrates the advantage of the proposed algorithm in comparison with state-of-the-art BSS methods. I. PROBLEM FORMULATION Suppose we are given m noisy observations {Xi}i=1..m of unknown linear mixtures of n ≤ m sparse sources {Sj}j=1..n with t > m samples. It is generally assumed that these data are corrupted by a Gaussian noise, accounting for instrumental or model imperfections. However in many applications, some entries are additionally corrupted by outliers, leading to the following model: X = AS + O + N, with X the observations, A the mixing matrix, S the sources, O the outliers, and N the Gaussian noise. The presence of outliers in real-world data have been taken into account in related domains such as compressive-sensing [9], dictionary learning [8] and denoising of hyperspectral images [4]. Contrary to these problems, both the mixing matrix and the sources should be individually retrieved in our problem, bringing further difficulties. In this case, the key issue lies in separating the components O from AS. To this end, assuming that the term AS has low-rank, some strategies [4] suggest to pre-process the data to estimate and remove the outliers with RPCA [3]. However, besides the fact that lowrankness is generally restrictive for most BSS problems, the source separation is severely hampered if the outliers are not well estimated. Therefore, we introduce a method that estimates the sources in the presence of the outliers without pre-processing. For the best of our knowledge, it has only been studied in [5] by using the β-divergence. Unlike [5], we propose to estimate jointly the outliers and the sources by exploiting their sparsity.