Measuring Distance between Reeb Graphs

We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions. As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs.

我们提出了一个用于REEB图的度量，称为功能失真距离。在此距离下，REEB图在输入函数的微小变化中是稳定的。同时，它在区分输入函数方面保持歧视性。特别是，主要的结果是，两个ReeB图之间的功能失真距离从下面通过普通持续图和扩展的持久图之间的瓶颈距离界定。作为结果的应用，我们分析了用于REEB图的自然简化方案，并表明Reeb图中的持久特征在简化中保持持久性。了解简化的Reeb图的重要特征的稳定性本身就是一个有趣的问题，对于Reeb图的实际用法至关重要。