A spectral gap precludes low-dimensional embeddings

Research output: Chapter in Book/Report/Conference proceedingConference contribution

8 Scopus citations


We prove that there is a universal constant C > 0 with the following property. Suppose that n ∈ ℕ and that A = (aij) ∈ Mn(ℝ) is a symmetric stochastic matrix. Denote the second-largest eigenvalue of A by λ2(A). Then for any finite-dimensional normed space (X, ⊥ · ⊥) we have (Equation presented) It follows that if an n-vertex O(1)-expander embeds with average distortion D ≧ 1 into X, then necessarily dim(X) ≳ nc/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) ≳ (log n)2/D2 of Linial, London and Rabinovich, strengthens a theorem of Matoušek, and answers a question of Andoni, Nikolov, Razenshteyn and Waingarten.

Original languageEnglish (US)
Title of host publication33rd International Symposium on Computational Geometry, SoCG 2017
EditorsMatthew J. Katz, Boris Aronov
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Number of pages4516
ISBN (Electronic)9783959770385
StatePublished - Jun 1 2017
Event33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia
Duration: Jul 4 2017Jul 7 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Other33rd International Symposium on Computational Geometry, SoCG 2017

All Science Journal Classification (ASJC) codes

  • Software


  • Complex interpolation
  • Dimensionality reduction
  • Expander graphs
  • Markov type
  • Metric embeddings
  • Nearest neighbor search
  • Nonlinear spectral gaps

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