发表机构
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究计算有限参数集的最小包围布雷格曼球问题，通过证明其与加权点集的幂距离MEBs等价，给出Frank-Wolfe $(1+\epsilon)$近似算法，还表明布雷格曼势提升变换可重新解释，揭示了布雷格曼MEB外心位置。
AI 中文摘要
在这项工作中，我们重新审视计算有限参数集的最小包围布雷格曼球（Bregman MEBs）的问题。首先，我们表明布雷格曼MEBs等同于相应加权点集关于幂距离的MEBs。然后，我们报告了一种高效的Frank-Wolfe $(1+\epsilon)$近似算法来计算幂MEBs，对于任意$\epsilon>0$。当在对偶梯度空间中表示时，此幂MEB近似算法与Nock和Nielsen（2005）的布雷格曼MEB近似算法一致。最后，我们表明用于构建布雷格曼Voronoi图的布雷格曼势提升变换可重新解释为应用于相应加权点集的经典抛物面提升变换。特别是，布雷格曼MEB外心位于最远的布雷格曼Voronoi图上，或者等效地位于相应的最远幂图上。
英文摘要
In this work, we revisit the problem of computing minimum enclosing Bregman balls (Bregman MEBs) of finite sets of parameters. First, we show that Bregman MEBs are equivalent to MEBs of corresponding weighted point sets with respect to the power distance. We then report an efficient Frank--Wolfe $(1+ε)$-approximation algorithm for computing power MEBs, for any $ε>0$. This power MEB approximation algorithm coincides with the Bregman MEB approximation algorithm of Nock and Nielsen (2005) when expressed in the dual gradient space. Finally, we show that the Bregman potential lifting transforms used to construct Bregman Voronoi diagrams can be reinterpreted as the classical paraboloid lifting transform applied to corresponding weighted point sets. In particular, Bregman MEB circumcenters lie on the farthest Bregman Voronoi diagrams or equivalently on the corresponding farthest power diagrams.
Comments39 pages, 22 figures, 1 table
