On the Support Recovery of Jointly Sparse Gaussian Sources using Sparse Bayesian Learning

Abstract—In this work, we provide non-asymptotic, probabilistic guarantees for successful sparse support recovery by the multiple sparse Bayesian learning (M-SBL) algorithm in the multiple measurement vector (MMV) framework. For joint sparse Gaussian sources, we show that M-SBL perfectly recovers their common nonzero support with arbitrarily high probability using only finitely many MMVs. In fact, the support error probability decays exponentially fast with the number of MMVs, with the decay rate depending on the restricted isometry property of the self Khatri-Rao product of the measurement matrix. Our analysis theoretically confirms that M-SBL is capable of recovering supports of size as high as O(m), where m is the number of measurements per sparse vector. In contrast, popular MMV algorithms in compressed sensing such as simultaneous orthogonal matching pursuit and row-LASSO can recover only O(m) sized supports. In the special case of noiseless measurements, we show that a single MMV suffices for perfect recovery of the k-sparse support in M-SBL, provided any k + 1 columns of the measurement matrix are linearly independent. Unlike existing support recovery guarantees for M-SBL, our sufficient conditions are non-asymptotic in nature, and do not require the orthogonality of the nonzero rows of the joint sparse signals.