Localization of More Sources Than Sensors via Jointly-Sparse Bayesian Learning

We analyze the jointly-sparse signal recovery problem in the regime where the number of sources k is larger than the number of measurements M. We show that the support set of sources can still be recovered with sparse Bayesian learning (M-SBL) even if k ≥ M. We provide sufficient conditions on the dictionary and sources which theoretically guarantee support set recovery in the noiseless case of M-SBL. We validate our sufficient conditions with experiments and also demonstrate that M-SBL outperforms M-CoSaMP, the algorithm recently used to localize more sources than sensors. Finally, we experimentally show robustness of the approach in the presence of noise.

[1]  Julia P. Owen,et al.  Robust Bayesian estimation of the location, orientation, and time course of multiple correlated neural sources using MEG , 2010, NeuroImage.

[2]  Vishal M. Patel.  Sparse and Redundant Representations for Inverse Problems and Recognition , 2010 .

[3]  R. Oostenveld,et al.  Independent EEG Sources Are Dipolar , 2012, PloS one.

[4]  Bhaskar D. Rao,et al.  An Empirical Bayesian Strategy for Solving the Simultaneous Sparse Approximation Problem , 2007, IEEE Transactions on Signal Processing.

[5]  Dustin G. Mixon,et al.  Certifying the Restricted Isometry Property is Hard , 2012, IEEE Transactions on Information Theory.

[6]  Karl J. Friston,et al.  Multiple sparse priors for the M/EEG inverse problem , 2008, NeuroImage.

[7]  Julia P. Owen,et al.  Estimating the Location and Orientation of Complex, Correlated Neural Activity using MEG , 2008, NIPS.

[8]  George Eastman House,et al.  Sparse Bayesian Learning and the Relevan e Ve tor Ma hine , 2001 .

[9]  Bhaskar D. Rao,et al.  Sparse solutions to linear inverse problems with multiple measurement vectors , 2005, IEEE Transactions on Signal Processing.

[10]  David P. Wipf,et al.  Bayesian methods for finding sparse representations , 2006 .

[11]  Jonathan Le Roux,et al.  Source localization in reverberant environments using sparse optimization , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[12]  Bhiksha Raj,et al.  Joint sparsity models for wideband array processing , 2011, Optical Engineering + Applications.

[13]  Michael Elad,et al.  Applications of Sparse Representation and Compressive Sensing , 2010, Proc. IEEE.

[14]  Alain Rakotomamonjy,et al.  Surveying and comparing simultaneous sparse approximation (or group-lasso) algorithms , 2011, Signal Process..