| Abstract: |
| We introduce a nonparametric way to estimate the global prob- ability 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 per- sistence 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 ac- curacy. Indeed, we prove that the associated kernel density estimate converges to the true distribution as the number of persistence di- agrams 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 ker- nel density estimation are presented for typical underlying datasets. |
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