Document # ε-Isometric Dimension Reduction for Incompressible Subsets of 𝓁_p

### Author Alexandros Eskenazis  ## File

LIPIcs.SoCG.2022.40.pdf
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## Acknowledgements

I am grateful to Keith Ball, Assaf Naor and Pierre Youssef for insightful discussions and useful feedback. I also wish to thank the anonymous referees for their constructive comments.

## Cite As

Alexandros Eskenazis. ε-Isometric Dimension Reduction for Incompressible Subsets of 𝓁_p. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 40:1-40:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.40

## 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 ∈ 𝓁_p^d such that ∀i,j∈{1,…,n}, ‖x_i-x_j‖^p_{L_p(μ)}-ε ≤ ‖y_i-y_j‖_{𝓁_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_{𝓁_p} does not admit ε-isometric dimension reduction by linear operators for any value of p≠2.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Random projections and metric embeddings
• Mathematics of computing → Probabilistic algorithms
• Mathematics of computing → Approximation
##### Keywords
• Dimension reduction
• ε-isometric embedding
• Maurey’s empirical method
• change of measure

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