G-Mapper: Learning a Cover in the Mapper Construction

The Mapper algorithm is a visualization technique in topological data analysis (TDA) that outputs a graph reflecting the structure of a given dataset. However, the Mapper algorithm requires tuning several parameters in order to generate a ``nice"Mapper graph. This paper focuses on selecting the cover parameter. We present an algorithm that optimizes the cover of a Mapper graph by splitting a cover repeatedly according to a statistical test for normality. Our algorithm is based on $G$-means clustering which searches for the optimal number of clusters in $k$-means by iteratively applying the Anderson-Darling test. Our splitting procedure employs a Gaussian mixture model to carefully choose the cover according to the distribution of the given data. Experiments for synthetic and real-world datasets demonstrate that our algorithm generates covers so that the Mapper graphs retain the essence of the datasets, while also running significantly fast.

Mapper算法是拓扑数据分析（TDA）中的一种可视化技术，它输出一个反映给定数据集结构的图。然而，Mapper算法需要调整几个参数才能生成一个“良好的”Mapper图。本文重点关注覆盖参数的选择。我们提出了一种算法，该算法通过根据正态性的统计检验反复拆分覆盖来优化Mapper图的覆盖。我们的算法基于$G$-均值聚类，它通过迭代应用安德森 - 达令检验在$k$-均值中搜索最优聚类数。我们的拆分过程采用高斯混合模型，根据给定数据的分布精心选择覆盖。对合成数据集和真实世界数据集的实验表明，我们的算法生成的覆盖使得Mapper图保留了数据集的本质，同时运行速度也非常快。