ADMM-based approach for compressive sensing with negative weights

In general, weighted compressive sensing recovery needs to solve optimisation problems with the objective function being the sum of a weighted l 1 -norm and a regularised differentiable convex function. Note that the weights in the weight vector are assumed to be positive. In fact, it is possible to achieve outstanding signal recovery performance even if some of the weights are appropriately designed to be negative. However, the negative weights lead to the non-convexity of the optimisation problem, which would place a barrier to attain the optimal solution. To deal with this issue, in this study, the authors propose to solve the problem by applying a well-established algorithm, namely, the alternating direction method of multipliers (ADMM) algorithm. It is shown that the optimal solution can be obtained by taking advantage of this optimisation scheme. The performance of the proposed algorithm is demonstrated by numerical results.