Multidimensional scaling (MDS) is a dimensionality reduction tool used for information analysis, data visualization and manifold learning. Most MDS procedures embed data points in low-dimensional euclidean (flat) domains, such that distances between the points are as close as possible to given inter-point dissimilarities. We present an efficient solver for classical scaling, a specific MDS model, by extrapolating the information provided by distances measured from a subset of the points to the remainder. The computational and space complexities of the new MDS methods are thereby reduced from quadratic to quasi-linear in the number of data points. Incorporating both local and global information about the data allows us to construct a low-rank approximation of the inter-geodesic distances between the data points. As a by-product, the proposed method allows for efficient computation of geodesic distances.

中文翻译：

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高效的测地距离计算和快速经典缩放


多维尺度（MDS）是一种用于信息分析、数据可视化和流形学习的降维工具。大多数 MDS 程序将数据点嵌入低维欧几里德（平坦）域中，使得点之间的距离尽可能接近给定的点间差异。我们通过推断从点的子集到其余点测量的距离提供的信息，提出了一种用于经典缩放的有效求解器，即特定的 MDS 模型。因此，新 MDS 方法的计算和空间复杂度从数据点数量的二次降低到准线性。结合数据的局部和全局信息使我们能够构造数据点之间测地距离的低秩近似。作为副产品，所提出的方法可以有效计算测地距离。