Nonparametric Estimation of Probability Density Functions of Random Persistence Diagrams
arXiv:1803.02739
Abstract
We introduce a nonparametric way to estimate the global probability density function for a random persistence diagram. Precisely, a kernel density function centered at a given persistence diagram and a given bandwidth is constructed. Our approach encapsulates the number of topological features and considers the appearance or disappearance of features near the diagonal in a stable fashion. In particular, the structure of our kernel individually tracks long persistence features, while considering features near the diagonal as a collective unit. The choice to describe short persistence features as a group reduces computation time while simultaneously retaining accuracy. Indeed, we prove that the associated kernel density estimate converges to the true distribution as the number of persistence diagrams increases and the bandwidth shrinks accordingly. We also establish the convergence of the mean absolute deviation estimate, defined according to the bottleneck metric. Lastly, examples of kernel density estimation are presented for typical underlying datasets.
arXiv comment39 pages, 14 figures
References in corpus (10)
- Confidence sets for persistence diagrams
- Persistence Images: A Stable Vector Representation of Persistent Homology
- Persistence weighted Gaussian kernel for topological data analysis
- Persistent Homology for Random Fields and Complexes
- Persistent Homology analysis of Phase Transitions
- Topological consistency via kernel estimation
- Parametric Inference using Persistence Diagrams: A Case Study in Population Genetics
- Modeling and replicating statistical topology, and evidence for CMB non-homogeneity
- Geometry Helps to Compare Persistence Diagrams
- A Stable Multi-Scale Kernel for Topological Machine Learning