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Multidimensional Scaling: Approximation and Complexity
Erik Demaine · Adam C Hesterberg · Frederic Koehler · Jayson Lynch · John C Urschel

Wed Jul 21 07:30 PM -- 07:35 PM (PDT) @
in Learning Theory 13 »

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.

Keywords: Computational Learning Theory

Author Information

Erik Demaine (MIT)
Adam C Hesterberg (Harvard John A. Paulson School Of Engineering And Applied Sciences)
Frederic Koehler (MIT)
Jayson Lynch (University of Waterloo)
John C Urschel (Massachusetts Institute of Technology)

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