| Abstract: |
| Persistence diagrams (PDs) are central objects in topological data analysis, capturing multiscale geometric structure in a stable and interpretable form. While statistical methods for analyzing collections of PDs are well developed, learning probability distributions on diagram space remains challenging due to variable cardinality and geometric constraints.
We propose a generative framework for persistence diagrams based on stochastic dynamics on diagram space. Specifically, we construct Markov processes driven by local edit operations---addition, relocation, and deletion of points---combined with projection mechanisms that enforce diagram validity. We show that the resulting process is irreducible, aperiodic, and geometrically ergodic, and therefore admits a unique stationary distribution.
To learn from data, we parameterize the transition dynamics via policies and design reward functions that approximate distributional discrepancies between the stationary distribution and empirical diagram distributions. This establishes a connection between policy optimization and distribution matching on non-Euclidean spaces.
Experiments on synthetic and real datasets demonstrate that the proposed approach captures key structural properties of diagram distributions, including persistence statistics and rare topological events. |
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