论文标题
通过内在距离重建歧管嵌入到欧几里得空间中
Reconstruction of manifold embeddings into Euclidean spaces via intrinsic distances
论文作者
论文摘要
我们考虑重建紧凑的连接的riemannian歧管在欧几里得空间中直至几乎等轴测的问题，鉴于有关距其“足够大”子集的固有距离的信息。这是经典的多种学习问题之一。碰巧的是，解决此类问题的最流行方法，具有悠久的数据科学历史，即经典的多维缩放（MDS）和最大差异（MVU）实际上错过了这一点，并且可能会提供远离等法的结果；此外，他们甚至可能没有嵌入Bi-Lipshitz。我们将简单地提供此问题的变分配方，这会导致算法总是提供几乎等轴测嵌入，而原始距离的变形如所需的较小（调节所需失真的上限的参数是该算法的输入参数）。
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).
