ε-Isometric Dimension Reduction for Incompressible Subsets of 𝓁_p

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.