UC San Diego

Sparse signal recovery and dictionary learning methods have found a vast number of applications including but not limited to data compression, machine learning and biomedical source localization/separation. The common underlying assumption in domains of application of these methods is that signals of interest are either sparse or can be sparsified in a transform domain. For source localization or identification this implies that the number of coefficients needed to represent the source signals in the transform domain should be less than the number of sensors. This evidently imposes constraints on the types of signals that can be recovered. In this work, we show that these constraints can be relaxed if the source signals are uncorrelated. Our work is inspired by the nature of electroencephalography (EEG) sources for which the independence assumption has been widely and successfully used. We focus on the multiple-measurement- vector (MMV) model of the sparse inverse problem. Under the assumption of uncorrelated sources, we first show that the required sparsity conditions for accurate signal support recovery can be relaxed which enables EEG source localization when more sources than sensors are simultaneously active. Later, we show that one can transform the traditional dictionary learning formulation into the covariance-domain to leverage the correlation information of the sources. Our covariance-domain dictionary learning framework can accurately identify the EEG scalp mixing matrix even when sources are not sparse in the traditional sense. This method enables the use of low-cost, low-density systems for high-density EEG brain imaging, which traditionally suffers from poor performance when using constraint-sensitive source separation algorithms like Independent Component Analysis. We also present locally-complete source separation algorithms that tackle the non-stationary nature of EEG sources. Finally, we present algorithms that targets identification of independent sources given an overcomplete dictionary. Our algorithms differ from the usual MMV sparse recovery algorithms in the sense that they optimize independence of the sources rather than their sparsity. We also a present robust bayesian algorithm for joint-sparse recovery in the MMV formulation