Multidimensional Scaling: Infinite Metric Measure Spaces
Multidimensional Scaling: Infinite Metric Measure Spaces
复制标题
多维尺度：无限度量测量空间
DOI：
--
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Lara Kassab
中科院分区：
文献类型：
--
作者：
Lara Kassab
多维标度（MDS）是一种将有限度量空间映射到低维欧氏空间的流行技术，其方式是最好地保持成对距离。研究了无限度量测度空间上的MDS的概念，沿着讨论了它的最优性和拟合优度。这使我们能够研究测地圆$S^1$到$mathbb{R}^m$的MDS嵌入，并提出关于测地$n$-球面$S^n$到$mathbb{R}^m$的MDS嵌入的问题。此外，我们解决的MDS的收敛问题。例如，如果一列度量测度空间收敛到一个固定的度量测度空间X，那么这些空间的MDS嵌入在什么意义上收敛到X的MDS嵌入？当序列中的每个度量空间具有相同的有限个点，或者当每个度量空间具有趋于无穷大的有限个点时，就可以理解收敛。我们还对序列中每个度量空间具有任意（可能是无限）点数时的收敛概念感兴趣。
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces, along with its optimality properties and goodness of fit. This allows us to study the MDS embeddings of the geodesic circle $S^1$ into $mathbb{R}^m$ for all $m$, and to ask questions about the MDS embeddings of the geodesic $n$-spheres $S^n$ into $mathbb{R}^m$. Furthermore, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space $X$, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of $X$? Convergence is understood when each metric space in the sequence has the same finite number of points, or when each metric space has a finite number of points tending to infinity. We are also interested in notions of convergence when each metric space in the sequence has an arbitrary (possibly infinite) number of points.
DOI：
--
发表时间：
2020
期刊：
ArXivorg
影响因子：
--
作者：
Blumstein, Mark;Kvinge, Henry
通讯作者：
Kvinge, Henry