ArXiv

Stochastic Dynamics on Persistence Diagram Space via Reinforcement Learning

2026-08-06 · 17:05 UTC ·Farzana Nasrin ·1 min read
Authors
Farzana Nasrin
Categories
stat.ML, cs.LG, math.AT
arXiv
https://arxiv.org/abs/2608.06276v1
PDF
https://arxiv.org/pdf/2608.06276v1

Brief

Nasrin develops an RL-based method for stochastic evolution of persistence diagrams that treats PDs as dynamic objects via local topology-aware edits. The approach yields controlled Markov chains on variable-cardinality PD spaces and proves key ergodic properties guaranteeing unique stationary distributions. Objectives target distribution matching, task-specific statistics, and compression; experiments on synthetic and neuroimaging PDs validate topology preservation with reduced complexity.

Why it matters

Farzana Nasrin (arXiv 2026-08-06) introduces a reinforcement-learning framework that implements topology-aware local edit operations to produce stochastic dynamics on persistence diagram (PD) space, defining controlled Markov processes on finite PDs with variable cardinality and proving irreducibility, aperiodicity, and geometric ergodicity (hence unique stationary laws).

Key details

  • The paper formulates reward objectives combining distribution matching, task-specific topological statistics, and structure-preserving compression (balancing distributional targets, diagram fidelity, and complexity reduction); experiments on synthetic and neuroimaging PDs show preservation of dominant topological features while reducing diagram complexity (27 pages, 7 figures, 5 tables).
Source evidence

Abstract

Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure. While substantial progress has been made in the statistical analysis of PDs, existing literature often treats diagrams as static objects and provide limited frameworks for probabilistic modeling and stochastic evolution on PD space. We introduce a reinforcement learning framework for stochastic dynamics on PD space, where diagrams evolve through topology aware local edit operations. The dynamics define controlled Markov processes on spaces of finite PDs with variable cardinality. We establish conditions under which the induced Markov chains are irreducible, aperiodic, and geometrically ergodic, implying the existence of unique stationary probability laws on PD space. To guide the dynamics toward scientifically relevant topological targets, we formulate objectives that encompass distribution matching, task specific topological statistics, and structure-preserving compression. The resulting rewards balance task specific distributional targets, diagram fidelity, and complexity reduction, and yield a framework for adaptive topological simplification and probabilistic modeling. Experiments on synthetic and neuroimaging PDs demonstrate that the proposed framework can preserve dominant topological structure while reducing diagram complexity.

Comment: 27 pages, 7 figures, and 5 tables