论文标题
无限的多维标度用于度量度量空间
Infinite multidimensional scaling for metric measure spaces
论文作者
论文摘要
对于给定的度量度量空间$（x，d，μ）$，我们考虑有限的点样本，计算它们之间的距离矩阵，然后使用多维缩放量表（MDS）算法将点在某些有限维空间中重建点，以此距离矩阵作为输入。我们表明，随着样品中的点的数量增长到无穷大，点的密度接近度量$μ$，因此此过程给出了自然限制。该限制可以看作是原始空间的“无限MD”嵌入，现在不再陷入有限维空间中，而是进入无限量化的希尔伯特空间。我们进一步表明，这种嵌入相对于度量空间的自然收敛是稳定的。但是，与通常在应用中通常认为的相反，我们表明，在许多情况下，它并不能保留距离，甚至是Bi-lipschitz，但可能会提供原始空间的雪花（Assouad-type）嵌入到Hilbert空间中（例如，这是一个球体的情况，是带有其地理底部的圆环的情况）。
For a given metric measure space $(X,d,μ)$ we consider finite samples of points, calculate the matrix of distances between them and then reconstruct the points in some finite-dimensional space using the multidimensional scaling (MDS) algorithm with this distance matrix as an input. We show that this procedure gives a natural limit as the number of points in the samples grows to infinity and the density of points approaches the measure $μ$. This limit can be viewed as "infinite MDS" embedding of the original space, now not anymore into a finite-dimensional space but rather into an infinitedimensional Hilbert space. We further show that this embedding is stable with respect to the natural convergence of metric measure spaces. However, contrary to what is usually believed in applications, we show that in many cases it does not preserve distances, nor is even bi-Lipschitz, but may provide snowflake (Assouad-type) embeddings of the original space to a Hilbert space (this is, for instance, the case of a sphere and a flat torus equipped with their geodesic distances).
