Metrics for Learning in Topological Persistence
This work addresses the need for robust topological descriptors in data analysis, particularly for applications like physical activity classification and atmospheric science, but it appears incremental as it builds on existing persistence methods.
The paper tackles the problem of stabilizing invariants in topological persistence analysis by defining metrics via contour functions, which are used as fingerprints for data classification. It demonstrates the approach by enhancing classification of physical activities data and providing robust descriptors for atmospheric cloud fields.
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can stabilize invariants characterizing these objects. We outline how so called contour functions induce relevant metrics for stabilizing the rank invariant. On the practical level, the stable ranks are used as fingerprints for data. Different choices of contour lead to different stable ranks and the topological learning is then the question of finding the optimal contour. We outline our analysis pipeline and show how it can enhance classification of physical activities data. As our main application we study how stable ranks and contours provide robust descriptors of spatial patterns of atmospheric cloud fields.