Elemental approximate message passing algorithm for sparse signal recovery

Vector approximate message passing (VAMP) has emerged as an effective and robust solution for sparse signal recovery (SSR). However, it could face a substantial computational burden when the dictionary matrix undergoes frequent variations in practical implementations. In this paper, we will illustrate that the challenges encountered by VAMP mainly arise from a matrix inversion operation. To circumvent this matrix inversion, we propose an elemental AMP-based algorithm by introducing additional auxiliary variables. This enables the processing of measurements element-by-element, thereby efficiently transforming any matrix operations into vector multiplications. Moreover, the proposed elemental AMP-based algorithm allows for adopting much more flexible approximation strategies (e.g., diagonal approximation) rather than resorting to the essential and overly simplistic coarse averaging operation as in VAMP. These innovations potentially contribute to both the reduction in computational complexity and improvement in recovery performance.