TY - GEN

T1 - A spectral gap precludes low-dimensional embeddings

AU - Naor, Assaf

N1 - Publisher Copyright:
© Assaf Naor.

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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.

AB - 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.

KW - Complex interpolation

KW - Dimensionality reduction

KW - Expander graphs

KW - Markov type

KW - Metric embeddings

KW - Nearest neighbor search

KW - Nonlinear spectral gaps

UR - http://www.scopus.com/inward/record.url?scp=85024373776&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85024373776&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2017.50

DO - 10.4230/LIPIcs.SoCG.2017.50

M3 - Conference contribution

AN - SCOPUS:85024373776

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 501

EP - 5016

BT - 33rd International Symposium on Computational Geometry, SoCG 2017

A2 - Katz, Matthew J.

A2 - Aronov, Boris

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 33rd International Symposium on Computational Geometry, SoCG 2017

Y2 - 4 July 2017 through 7 July 2017

ER -