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.
@InProceedings{naor:LIPIcs.SoCG.2017.50, author = {Naor, Assaf}, title = {{A Spectral Gap Precludes Low-Dimensional Embeddings}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {50:1--50:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.50}, URN = {urn:nbn:de:0030-drops-71822}, doi = {10.4230/LIPIcs.SoCG.2017.50}, annote = {Keywords: Metric embeddings, dimensionality reduction, expander graphs, nonlinear spectral gaps, nearest neighbor search, complex interpolation, Markov type.} }
Feedback for Dagstuhl Publishing