论文标题
对广义的坎托维奇 - 罗宾斯坦规范的极端点和稀疏优化
Extremal points and sparse optimization for generalized Kantorovich-Rubinstein norms
论文作者
论文摘要
非平滑惩罚的超级级别集合的极端点的精确表征既可以提供有关最小化器的详细信息，又提供了涉及它们的最小化问题的一般类别中的最佳条件。此外，它可以为其有效的解决方案应用加速的通用条件梯度方法。在此手稿中，该程序适应了平滑凸足项的最低限制，该凸足术语以不平衡的运输正规化项增强，以广义的Kantorovich-Rubinstein Norm的形式进行ra仪。更确切地说，我们表明，与后者相关的极端点是由空间域中支持的所有Dirac Delta功能以及某些偶极子（即具有相同质量但具有不同符号的Diracs对）给出的。随后，该表征用于得出精确的一阶最优条件以及在自然假设下证明线性收敛的有效解决方案算法。此行为也反映在模型问题的数值示例中。
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.
