$$\varepsilon $$-Isometric Dimension Reduction for Incompressible Subsets of $$\ell _p$$

Abstract

AbstractFix $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) , $$K\in (0,\infty )$$ K ∈ ( 0 , ∞ ) , and a probability measure $$\mu $$ μ . We prove that for every $$n\in \mathbb {N}$$ n ∈ N , $$\varepsilon \in (0,1)$$ ε ∈ ( 0 , 1 ) , and $$x_1,\ldots ,x_n\in L_p(\mu )$$ x 1 , … , x n ∈ L p ( μ ) with $$\big \Vert \max _{i\in \{1,\ldots ,n\}} |x_i| \big \Vert _{L_p(\mu )} \le K$$ ‖ max i ∈ { 1 , … , n } | x i | ‖ L p ( μ ) ≤ K , there exist $$d\le \frac{32e^2 (2K)^{2p}\log n}{\varepsilon ^2}$$ d ≤ 32 e 2 ( 2 K ) 2 p log n ε 2 and vectors $$y_1,\ldots , y_n \in \ell _p^d$$ y 1 , … , y n ∈ ℓ p d such that $$\begin{aligned} {\forall }\,\,i,j\in \{1,\ldots ,n\}, \quad \Vert x_i-x_j\Vert ^p_{L_p(\mu )}-\varepsilon\le & {} \Vert y_i-y_j\Vert _{\ell _p^d}^p\le \Vert x_i-x_j\Vert ^p_{L_p(\mu )}+\varepsilon . \end{aligned}$$ ∀ i , j ∈ { 1 , … , n } , ‖ x i - x j ‖ L p ( μ ) p - ε ≤ ‖ y i - y j ‖ ℓ p d p ≤ ‖ x i - x j ‖ L p ( μ ) p + ε . Moreover, the argument implies the existence of a greedy algorithm which outputs $$\{y_i\}_{i=1}^n$$ { y i } i = 1 n after receiving $$\{x_i\}_{i=1}^n$$ { 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 $$\varepsilon $$ ε -isometric dimension reduction of the unit ball $${\textbf {B}}_E$$ B E of a normed space $$(E,\Vert \cdot \Vert _E)$$ ( E , ‖ · ‖ E ) and we prove that $${\textbf {B}}_{\ell _p}$$ B ℓ p does not admit $$\varepsilon $$ ε -isometric dimension reduction by linear operators for any value of $$p\ne 2$$ p ≠ 2 .