External Division of Two Proximity Operators: An Application to Signal Recovery with Structured Sparsity

This paper studies the external division operator, an external division (an affine combination with positive and negative weights) of two proximity operators. We show that the external division operator is cocoercive under some condition, and it can be expressed as the proximity operator of a certain weakly convex function. We then consider using the external division operator as an alternative to the proximity operator in the proximal gradient algorithm, which we show converges to a minimizer of the cost function penalized by the weakly convex function under some conditions. Our analysis covers the case when the fidelity function is convex but not strongly convex. In simulations, we employ the octagonal shrinkage and clustering algorithm for regression in the external division operator, and we show that significant improvements are attained in signal recovery with structured sparsity in both overdetermined and underdetermined cases.