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Gohberg-Semencul Factorization-Based Fast Implementation of Sparse Bayesian Learning With a Fourier Dictionary

Sparse Bayesian learning (SBL) is a popular and robust algorithm for sparse signal reconstruction (SSR). Unfortunately, the SBL algorithm suffers from heavy computational complexity when it is implemented directly since the inversion and multiplying operations of the estimation of the covariance matrix are involved in each iteration, which is proportional to the cube of the observed data length and thus prevents it solving large-scale problems. In many applications, such as radar imaging and array signal processing, the signal to be recovered is sparse in the Fourier dictionary. In this article, we propose an efficient implementation method for the Fourier dictionary-based SBL (FD-SBL). In the case that the Fourier dictionary is adopted, the estimation of the covariance matrix is a Toeplitz matrix for 1-D data or a Toeplitz-block-Toeplitz (TBT) matrix for 2-D data during the FD-SBL iterations. By utilizing this property, we employ the Gohberg–Semencul (G-S)-type factorization to accelerate the implementation of FD-SBL. To be noted, there is no approximation in our proposed method, and the computational cost is reduced by several orders of magnitude compared with the direct implementation of SBL. Finally, the experimental results verify the effectiveness of the proposed method.