用多图流学习复杂拓扑数据的几何结构
Learning Geometry and Topology via Multi-Chart Flows
- 设计多图流联合训练方案，突破单一流对非平凡拓扑的建模限制
- 首次实现基于多图流的测地线数值计算，提升拓扑估计精度
- 适合研究流形学习、拓扑数据分析的学者参考
真实世界数据常位于高维空间中的低维黎曼流形上。这促使我们学习退化的归一化流，将高维空间映射到低维潜在空间。然而，若流形具有非平凡拓扑，则单一流无法准确建模，必须使用多个流“拼接”而成。本文首先提出该类流集合的通用训练方案，其次开发了首个在多图流框架下计算测地线的数值算法。实验表明，该方法显著提升了拓扑估计性能。
原文摘要 · Abstract (English)
Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if the manifold has a non-trivial topology, it can never be correctly learned using a single flow. Instead multiple flows must be `glued together'. In this paper, we first propose the general training scheme for learning such a collection of flows, and secondly we develop the first numerical algorithms for computing geodesics on such manifolds. Empirically, we demonstrate that this leads to highly significant improvements in topology estimation.
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