用拓扑连通性分析数据流形，提升生成模型的稳定性和质量。
Manifold Percolation: from generative model to Reinforce learning
- 基于连续渗流理论构建流形支持分析新方法
- 提出可微损失函数，有效抑制模式崩溃与流形收缩
- 适合关注生成模型稳定性与高质量生成的研究者
生成建模通常被视为学习映射规则，但从无法访问这些规则的观察者视角来看，任务变为从概率分布中解耦几何支撑。我们提出，连续渗流在支撑分析中具有独特优势，因为采样过程将高维密度估计转化为对支撑上的几何计数问题。本文建立了随机几何图的拓扑相变与高维空间中底层数据流形之间的严格对应关系。通过分析所提出的渗流偏移（Percolation Shift）度量与FID之间的关系，我们发现该度量能捕捉标准统计指标失效的结构性缺陷，如隐式模式崩溃。最后，我们将这一拓扑现象转化为可微损失函数以指导训练。实验结果表明，该方法不仅能防止流形收缩，还能促进一种协同改进：拓扑稳定性成为静态生成与序列决策中持续高保真度的前提。
原文摘要 · Abstract (English)
Generative modeling is typically framed as learning mapping rules, but from an observer's perspective without access to these rules, the task becomes disentangling the geometric support from the probability distribution. We propose that continuum percolation is uniquely suited to this support analysis, as the sampling process effectively projects high-dimensional density estimation onto a geometric counting problem on the support. In this work, we establish a rigorous correspondence between the topological phase transitions of random geometric graphs and the underlying data manifold in high-dimensional space. By analyzing the relationship between our proposed Percolation Shift metric and FID, we show that this metric captures structural pathologies, such as implicit mode collapse, where standard statistical metrics fail. Finally, we translate this topological phenomenon into a differentiable loss function that guides training. Experimental results confirm that this approach not only prevents manifold shrinkage but also fosters a form of synergistic improvement, where topological stability becomes a prerequisite for sustained high fidelity in both static generation and sequential decision making.
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