Extremal points and sparse optimization for generalized Kantorovich-Rubinstein norms

TL;DR

This paper characterizes extremal points of generalized Kantorovich-Rubinstein norms, enabling efficient optimization algorithms for unbalanced transport regularization in convex problems, with proven linear convergence and numerical validation.

Contribution

It provides a detailed extremal point characterization for generalized Kantorovich-Rubinstein norms, leading to new optimality conditions and an efficient, convergent solution algorithm.

Findings

Extremal points are Dirac deltas and dipoles.

Derived first-order optimality conditions.

Proved linear convergence of the algorithm.

Abstract

A precise characterization of the extremal points of sublevel sets of nonsmooth penalties provides both detailed information about minimizers, and optimality conditions in general classes of minimization problems involving them. Moreover, it enables the application of accelerated generalized conditional gradient methods for their efficient solution. In this manuscript, this program is adapted to the minimization of a smooth convex fidelity term which is augmented with an unbalanced transport regularization term given in the form of a generalized Kantorovich-Rubinstein norm for Radon measures. More precisely, we show that the extremal points associated to the latter are given by all Dirac delta functionals supported in the spatial domain as well as certain dipoles, i.e., pairs of Diracs with the same mass but with different signs. Subsequently, this characterization is used to derive precise first-order optimality conditions as well as an efficient solution algorithm for which linear convergence is proved under natural assumptions. This behaviour is also reflected in numerical examples for a model problem.