将流形学习扩展到任意度量空间，让非欧距离也能有效降维。
Manifold learning in metric spaces
- 用度量空间替代欧氏空间，推广图拉普拉斯方法
- 证明特定度量下图拉普拉斯可点收敛
- 适用于水琴斯坦等非欧距离场景
基于拉普拉斯的方法在高维数据降维中广泛应用，其理论依赖于欧氏距离能局部逼近数据所在流形的测地距离。然而，在某些应用中，如水琴斯坦距离，其他度量可能更合适。本文提出一种将流形学习推广至度量空间的框架，并研究何种度量能满足图拉普拉斯点收敛的充分条件。
原文摘要 · Abstract (English)
Laplacian-based methods are popular for the dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance on the underlying submanifold which the data are assumed to lie on. However, for some applications, other metrics, such as the Wasserstein distance, may provide a more appropriate notion of distance than the Euclidean distance. We provide a framework that generalizes the problem of manifold learning to metric spaces and study when a metric satisfies sufficient conditions for the pointwise convergence of the graph Laplacian.
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