Mapper–Type Algorithms for Complex Data and Relations

Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar–valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to Ball Mapper that enable encoding of the structure, internal relations and symmetries of the point cloud. Moreover, the strengths of Mapper and Ball Mapper constructions are combined to create a tool for comparing high dimensional data descriptors of a single dataset. This new hybrid algorithm, Mapper on Ball Mapper, is applicable to high dimensional lens functions. As a proof of concept we include applications to knot and game theory.

Mapper与Ball Mapper均为拓扑数据分析（Topological Data Analysis）工具，可用于探索高维点云，并对这些点云上的标量值函数进行可视化。受纽结理论中悬而未决问题的启发，研究人员为Ball Mapper新增了若干功能，使其能够对点云的结构、内部关联与对称性进行编码。此外，研究人员结合Mapper与Ball Mapper两种算法的优势，开发出一款可用于对比单个数据集的高维数据描述符的工具。这款全新的混合算法——Ball Mapper上的Mapper（Mapper on Ball Mapper），可适用于高维透镜函数。作为概念验证，本数据集包含了该算法在纽结理论与博弈论中的应用案例。