通过非收缩流学习数据流形的全局坐标系，实现可解释的几何建模。
Discovering Data Manifold Geometry via Non-Contracting Flows
- 在高维空间中学习切向量场，构建全局参考框架。
- 在合成数据上实现正确切线对齐和一致坐标结构。
- 适用于需要几何感知表示的下游任务，如分类与可视化。
我们提出一种无监督方法，通过在原始空间中学习切向量场来构建全局参考系统，这些向量场覆盖未知数据流形的切空间。与隐含流形平坦性的等距目标不同，该方法学习的流将所有样本传输至一个可学习的公共参考点，沿流的弧长定义了与共享全局框架关联的可解释内蕴坐标。为防止退化坍缩，我们引入非收缩约束，并推导出一种无需积分、受流匹配启发的可扩展目标函数。在理论框架下，我们证明最小化该目标能恢复全局坐标图谱（当其存在时）。实验表明，该方法在合成流形上实现了正确的切线对齐和连贯的全局坐标结构；在CIFAR-10上也展现出良好可扩展性，所学坐标在下游分类任务中达到竞争性性能。
原文摘要 · Abstract (English)
We introduce an unsupervised approach for constructing a global reference system by learning, in the ambient space, vector fields that span the tangent spaces of an unknown data manifold. In contrast to isometric objectives, which implicitly assume manifold flatness, our method learns tangent vector fields whose flows transport all samples to a common, learnable reference point. The resulting arc-lengths along these flows define interpretable intrinsic coordinates tied to a shared global frame. To prevent degenerate collapse, we enforce a non-shrinking constraint and derive a scalable, integration-free objective inspired by flow matching. Within our theoretical framework, we prove that minimizing the proposed objective recovers a global coordinate chart when one exists. Empirically, we obtain correct tangent alignment and coherent global coordinate structure on synthetic manifolds. We also demonstrate the scalability of our method on CIFAR-10, where the learned coordinates achieve competitive downstream classification performance.
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