发表机构
University of Tennessee（田纳西大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种用于持续同调图（PD）空间随机动力学的强化学习框架，建立了相关马尔可夫链的遍历条件，可实现自适应拓扑简化与概率建模，实验验证其能保留主导拓扑结构并降低图复杂度。
AI 中文摘要
持续同调图（PDs）为多尺度拓扑结构提供了稳定且可解释的摘要。尽管PDs的统计分析已取得显著进展，但现有文献常将图视为静态对象，针对PD空间上的概率建模与随机演化的框架十分有限。本文提出一种用于PD空间随机动力学的强化学习框架，其中图通过拓扑感知的局部编辑操作演化。该动力学在具有可变基数的有限PD空间上定义了受控马尔可夫过程。本文建立了诱导马尔可夫链不可约、非周期且几何遍历的条件，表明PD空间上存在唯一的平稳概率律。为引导动力学向科学相关的拓扑目标演化，本文构建了包含分布匹配、任务特定拓扑统计及结构保持压缩的目标。所得奖励平衡了任务特定分布目标、图保真度与复杂度降低，形成了自适应拓扑简化与概率建模的框架。在合成数据及神经成像PD上的实验表明，该框架可在降低图复杂度的同时保留主导拓扑结构。
英文摘要
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.
Comments27 pages, 7 figures, and 5 tables
