Metric Multidimensional scaling (MDS) is a classical method for generating meaningful (non-linear) low-dimensional embeddings of high-dimensional data. MDS has a long history in the statistics, machine learning, and graph drawing communities. In particular, the Kamada-Kawai force-directed graph drawing method is equivalent to MDS and is one of the most popular ways in practice to embed graphs into low dimensions. Despite its ubiquity, our theoretical understanding of MDS remains limited as its objective function is highly non-convex. In this paper, we prove that minimizing the Kamada-Kawai objective is NP-hard and give a provable approximation algorithm for optimizing it, which in particular is a PTAS on low-diameter graphs. We supplement this result with experiments suggesting possible connections between our greedy approximation algorithm and gradient-based methods.

翻译：MDS是生成有意义(非线性)高维数据低维嵌入的经典方法。 MDS在统计、机器学习和绘图界有着悠久的历史。 特别是, Kamada-Kawai 强制引导的图形绘制方法相当于MDS, 是将图形嵌入低维的最常用方法之一 。 尽管它无处不在, 我们对MDDS的理论理解仍然有限, 因为它的客观功能是高度非连接的。 在本文中, 我们证明, 尽可能减少 Kamada- Kawai 目标是坚固的, 并为优化它提供了一种可行的近似算法, 特别是低直径图上的 PTAS 。 我们用实验来补充这一结果, 表明我们贪婪近似算法和梯度方法之间可能存在的联系 。