One of the prevailing ideas in geometric and topological data analysis is to provide descriptors that encode useful information about hidden objects from observed data. The Reeb graph is one such descriptor for a given scalar function. The Reeb graph provides a simple yet meaningful abstraction of the input domain, and can also be computed efficiently. Given the popularity of the Reeb graph in applications, it is important to understand its stability and robustness with respect to changes in the input function, as well as to be able to compare the Reeb graphs resulting from different functions. In this paper, we propose a metric for Reeb graphs, called the functional distortion distance. Under this distance measure, 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 (and thus more discriminative than) 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.
- Pub Date:
- July 2013
- Computer Science - Computational Geometry;
- Mathematics - Algebraic Topology;
- Mathematics - Metric Geometry
- Updated to match the modified definitions of the conference version. Improved and simplified bounds for the Gromov-Hausdorff distance. New section on interleaving distance for merge trees. Further changes and improvements throughout the paper