Title: 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 $\mathcal{O}(m^2)$, 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
$\mathcal{O}(m)$ 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.