Erik Demaine, Adam Hesterberg, Frederic Koehler, Jayson Lynch, and John Urschel, “Multidimensional Scaling: Approximation and Complexity”, in Proceedings of the 38th International Conference on Machine Learning (ICML 2021), edited by Meila, Marina and Zhang, Tong, Proceedings of Machine Learning Research, volume 139, July 18–24, 2021, pages 2568–2578.
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.