An Approximated ADMM based Algorithm for $\ell_{1}-\ell_{2}$ Optimization Problem

Compressed sensing is a technique to recover a sparse vector from its underdetermined linear measurements. Since a naive $\ell_{0}$ optimization approach is hard to tackle due to the discreteness and the non-convexity of $\ell_{0}$ norm, a relaxed problem of the $\ell_{1}-\ell_{2}$ optimization is often employed for the reconstruction of the sparse vector especially when the measurement noise is not negligible. FISTA (fast iterative shrinkage-thresholding algorithm) is one of popular algorithms for the $\ell_{1}-\ell_{2}$ optimization, and is known to achieve optimal convergence rate among the first order methods. Recently, the employment of optical circuits for various signal processing including deep neural networks has been considered intensively, but it is difficult to implement FISTA with the optical circuit, because it requires operations of divisions with a dynamic value in the algorithm. In this paper, assuming the implementation with the optical circuit, we propose an ADMM (alternating direction method of multipliers) based algorithm for the $\ell_{1}-\ell_{2}$ optimization. It is true that an ADMM based algorithm for the $\ell_{1}-\ell_{2}$ optimization has been already proposed in the literature, but the proposed algorithm is derived with the different formulation from the existing method, and unlike the existing ADMM based algorithm, the proposed algorithm does not include the calculation of the inverse of a matrix. Computer simulation results demonstrate that the proposed algorithm can achieve comparable performance as FISTA or existing ADMM based algorithm while requiring no division operations and no matrix inversions.