License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.40
URN: urn:nbn:de:0030-drops-160486
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16048/
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Eskenazis, Alexandros

ε-Isometric Dimension Reduction for Incompressible Subsets of ?_p

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LIPIcs-SoCG-2022-40.pdf (0.8 MB)

Abstract

Fix p ∈ [1,∞), K ∈ (0,∞) and a probability measure μ. We prove that for every n ∈ ℕ, ε ∈ (0,1) and x₁,…,x_n ∈ L_p(μ) with ‖max_{i ∈ {1,…,n}}|x_i|‖_{L_p(μ)} ≤ K, there exists d ≤ (32e² (2K)^{2p}log n)/ε² and vectors y₁,…, y_n ∈ ?_p^d such that
∀i,j∈{1,…,n}, ‖x_i-x_j‖^p_{L_p(μ)}-ε ≤ ‖y_i-y_j‖_{?_p^d}^p ≤ ‖x_i-x_j‖^p_{L_p(μ)}+ε.
Moreover, the argument implies the existence of a greedy algorithm which outputs {y_i}_{i = 1}ⁿ after receiving {x_i}_{i = 1}ⁿ as input. The proof relies on a derandomized version of Maurey’s empirical method (1981) combined with a combinatorial idea of Ball (1990) and a suitable change of measure. Motivated by the above embedding, we introduce the notion of ε-isometric dimension reduction of the unit ball B_E of a normed space (E,‖⋅‖_E) and we prove that B_{?_p} does not admit ε-isometric dimension reduction by linear operators for any value of p≠2.

BibTeX - Entry

@InProceedings{eskenazis:LIPIcs.SoCG.2022.40,
  author =	{Eskenazis, Alexandros},
  title =	{{\epsilon-Isometric Dimension Reduction for Incompressible Subsets of ?\underlinep}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{40:1--40:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16048},
  URN =		{urn:nbn:de:0030-drops-160486},
  doi =		{10.4230/LIPIcs.SoCG.2022.40},
  annote =	{Keywords: Dimension reduction, \epsilon-isometric embedding, Maurey’s empirical method, change of measure}
}

Keywords: Dimension reduction, ε-isometric embedding, Maurey’s empirical method, change of measure
Collection: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue Date: 2022
Date of publication: 01.06.2022

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