Metrics for Learning in Topological Persistence

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.

Topological descriptors of spatial coherence in a convective boundary layer

The interaction between a turbulent convective boundary layer (CBL) and the underlying land surface is an important research problem in the geosciences. In order to model this interaction adequately, it is necessary to develop tools which can describe it quantitatively. Commonly employed methods, such as bulk flow statistics, are known to be insufficient for this task, especially when land surfaces with equal aggregate statistics but different spatial patterns are involved. While geometrical properties of the surface forcing have a strong influence on flow structure, it is precisely those properties that get neglected when computing bulk statistics. Here, we present a set of descriptors based on low-level topological information (i.\,e. connectivity), and show how these can be used both in the structural analysis of the CBL and in modeling its response to differences in surface forcing. The topological property of connectivity is not only easier to compute than its higher-dimensional homological counterparts, but also has a natural relation to the physical concept of a coherent structure.