The persistence landscape and some of its properties
This work provides a method for applying machine learning to topological data analysis, which is incremental as it builds on existing persistence diagram techniques.
The paper tackles the problem of analyzing topological data via persistence diagrams by introducing persistence landscapes, which map these diagrams into function spaces, enabling the application of statistical and machine learning tools. It demonstrates that this mapping is stable and invertible, and introduces a weighted version with a one-parameter family of kernels for learning.
Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.