An Approximated ADMM based Algorithm for ℓ1−ℓ2 Optimization Problem

RUI LIN, Kazunori Hayashi

Abstract: Compressed sensing is a technique to recover asparse vector from its underdetermined linear measurements.Since a naive ℓ0 optimization approach is hard to tackle due to thediscreteness and the non-convexity of ℓ0 norm, a relaxed problemof the ℓ1−ℓ2 optimization is often employed for the reconstructionof the sparse vector especially when the measurement noise isnot negligible. FISTA (fast iterative shrinkage-thresholding algorithm) is one of popular algorithms for the ℓ1 − ℓ2 optimization,and is known to achieve optimal convergence rate among thefirst order methods. Recently, the employment of optical circuitsfor various signal processing including deep neural networks hasbeen considered intensively, but it is difficult to implement FISTAwith the optical circuit, because it requires operations of divisionswith a dynamic value in the algorithm. In this paper, assuming theimplementation with the optical circuit, we propose an ADMM(alternating direction method of multipliers) based algorithmfor the ℓ1 − ℓ2 optimization. It is true that an ADMM basedalgorithm for the ℓ1 −ℓ2 optimization has been already proposedin the literature, but the proposed algorithm is derived with thedifferent formulation from the existing method, and unlike theexisting ADMM based algorithm, the proposed algorithm doesnot include the calculation of the inverse of a matrix. Computersimulation results demonstrate that the proposed algorithm canachieve comparable performance as FISTA or existing ADMMbased algorithm while requiring no division operations and nomatrix inversions.