Persistence diagrams are common objects in the field of Topological Data Analysis. They are topological summaries that capture both topological and geometric structure within data. Recently there has been a surge of interest in developing tools to statistically analyse populations of persistence diagrams, a process hampered by the complicated geometry of the space of persistence diagrams. In this paper we study the median of a set of diagrams, defined as the minimizer of an appropriate cost function analogous to the sum of distances used for samples of real numbers. We then characterize the local minima of this cost function and in doing so characterize the median. We also do some comparative analysis of the properties of the median and the mean.

中文翻译：

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持久性图种群的中位数

持久图是拓扑数据分析领域的常见对象。它们是捕获数据中的拓扑结构和几何结构的拓扑摘要。最近，人们对开发用于统计分析持久图总体的工具产生了浓厚的兴趣，这一过程受到持久图空间复杂几何形状的阻碍。在本文中，我们研究了一组图的中值，定义为一个合适的成本函数的最小值，类似于用于实数样本的距离总和。然后我们表征这个成本函数的局部最小值，并以此表征中值。我们还对中位数和均值的属性进行了一些比较分析。