aiqbench analysis
This research introduces a reinforcement learning framework for modeling stochastic dynamics on persistence diagram spaces, enabling controlled Markov processes that preserve topological structure while reducing complexity. By formulating objectives around distribution matching and fidelity, the method offers new tools for probabilistic modeling in applications like neuroimaging where multiscale data analysis is critical. The work advances the intersection of algebraic topology and machine learning by treating diagrams as evolving systems rather than static summaries.
Stochastic Dynamics on Persistence Diagram Space via Reinforcement Learning
arXiv:2608.06276v1 Announce Type: cross 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.
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.
Machine Learning (stat.ML); Machine Learning (cs.LG); Algebraic Topology (math.AT)
arXiv:2608.06276 [stat.ML]
(or arXiv:2608.06276v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2608.06276 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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