Fast expectation propagation for sparse signal reconstruction with a fourier dictionary

Sparse signal reconstruction (SSR) involves tackling large underdetermined systems of linear equations while incorporating constraints or regularizers. Expectation propagation (EP) emerges as a robust method for SSR, converting these constraints into prior information. However, the cubic complexity of matrix inversion per EP cycle hinders its implementation in large systems without approximation. In various applications like direction of arrival estimation (DoA), radar imaging etc., the signal to be recovered exhibits sparsity in the Fourier dictionary. To address this, we present a fast EP algorithm based on the Gohberg-Semencul (G-S) formula and Levinson-Durbin (L-D) type algorithm, boasting only quadratic complexity. Notably, no approximation operations or random measurement matrices are required for matrix inversion compared to approximate message passing (AMP) and other message passing based algorithms. Furthermore, it is compatible with non-identically and independently distributed (n.i.i.d.) priors. Numerical simulations conclusively demonstrate the efficacy of fast EP.