通过内蕴距离将流形嵌入重构到欧几里得空间
Reconstruction of manifold embeddings into Euclidean spaces via intrinsic distances
AI总结：
针对由内蕴距离重构流形欧几里得嵌入的问题，本文指出 MDS 与 MVU 可能失效，并提出一种变分算法，可保证生成扭曲任意小的近似等距嵌入。
摘要：
我们考虑如下问题：在给定一个紧致连通 Riemann 流形的“足够大”子集上各点之间内蕴距离信息的条件下，将该流形在欧几里得空间中的嵌入重构到近似等距的程度。这是经典的流形学习问题之一。事实表明，处理这一问题的最流行方法——即在数据科学中具有悠久历史的经典多维标度（MDS）和最大方差展开（MVU）——实际上并未抓住问题关键，可能给出与等距相距甚远的结果；此外，它们甚至可能无法给出双 Lipschitz 嵌入。我们将给出该问题的一个简单变分表述，由此得到一种算法，该算法总能给出近似等距嵌入，且原始距离的扭曲可达到任意小（用于调节所需扭曲上界的参数是该算法的一个输入参数）。
英文摘要：
We consider the problem of reconstructing an embedding of a compact connected Riemannian manifold in a Euclidean space up to an almost isometry, given the information on intrinsic distances between points from its ``sufficiently large'' subset. This is one of the classical manifold learning problems. It happens that the most popular methods to deal with such a problem, with a long history in data science, namely, the classical Multidimensional scaling (MDS) and the Maximum variance unfolding (MVU), actually miss the point and may provide results very far from an isometry; moreover, they may even give no bi-Lipshitz embedding. We will provide an easy variational formulation of this problem, which leads to an algorithm always providing an almost isometric embedding with the distortion of original distances as small as desired (the parameter regulating the upper bound for the desired distortion is an input parameter of this algorithm).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
