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A Spectral Gap Precludes Low-Dimensional Embeddings

Author Assaf Naor

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Assaf Naor

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Assaf Naor. A Spectral Gap Precludes Low-Dimensional Embeddings. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 50:1-50:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.SoCG.2017.50

Abstract

We prove that if an n-vertex O(1)-expander embeds with average distortion D into a finite dimensional normed space X, then necessarily the dimension of X is at least n^{c/D} for some universal constant c>0. This is sharp up to the value of the constant c, and it improves over the previously best-known estimate dim(X)> c(log n)^2/D^2 of Linial, London and Rabinovich, strengthens a theorem of Matousek, and answers a question of Andoni, Nikolov, Razenshteyn and Waingarten.
Keywords
  • Metric embeddings
  • dimensionality reduction
  • expander graphs
  • nonlinear spectral gaps
  • nearest neighbor search
  • complex interpolation
  • Markov type.

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