Asymptotic Behavior of Super-Resolution Sparse Bayesian Learning

Sparse Bayesian Learning with dictionary refinement (SBL-DR) is a gridless technique for sparse signal reconstruction, focusing on super-resolution estimation of spectral line locations and their quantity. Its cost function coinsides with that of stochastic maximum likelihood (SML), a well-known method in array processing for estimating frequencies of complex exponentials. While SML exhibits consistency and efficiency with growing array snapshots or size, SBL-DR faces inconsistency with only one measurement snapshot. This study explores SBL-DR asymptotic behavior using a single measurement snapshot and growing sample size using Γ-convergence theory. It computes upper and lower bounds for the SBL-DR cost function, showing their convergence to a Γ-limit that is minimized at true signal locations. By leveraging the properties of Γ-convergence, it is established that the minima of the SBL-DR cost function asymptotically approach those of the Γ-limit function, thereby achieving consistency for SBL-DR.