External Division of Two Proximity Operators—Part II: Generalization and Properties

This paper consolidates the notion of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">external division operator</i>, an affine combination of two proximity operators with positive and negative weights. We show that the external division operator for two proximity operators of convex functions is a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">monotone Lipschitz gradient (MoL-Grad) denoiser</i> (the gradient of a smooth convex function, under some conditions). This immediately indicates that the external division operator can be expressed as the proximity operator of a certain weakly convex function. This allows the convergence analysis of the gradient algorithm based on the external division operator. Finally, it is shown that the external division operator is a generalization of the proximity operator of the Moreau enhancement when applied to the octagonal shrinkage and clustering algorithm for regression (OSCAR) regularizers. In Part I of this work, numerical examples show that the external division operator significantly outperforms the OSCAR regularizer in the application of feature grouping.