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Computer Science > Machine Learning

arXiv:2109.11505 (cs)
[Submitted on 23 Sep 2021]

Title:Multidimensional Scaling: Approximation and Complexity

Authors:Erik Demaine, Adam Hesterberg, Frederic Koehler, Jayson Lynch, John Urschel
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Abstract: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.
Subjects: Machine Learning (cs.LG); Computational Geometry (cs.CG); Data Structures and Algorithms (cs.DS); Machine Learning (stat.ML)
Cite as: arXiv:2109.11505 [cs.LG]
  (or arXiv:2109.11505v1 [cs.LG] for this version)
  https://doi.org/10.48550/arXiv.2109.11505

Submission history

From: John C Urschel [view email]
[v1] Thu, 23 Sep 2021 17:14:25 UTC (4,025 KB)
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