ε-isometric dimension reduction for incompressible subsets of ℓ_p
share
Fix p∈[1,∞), K∈(0,∞) and a probability measure μ. We prove that for every n∈ℕ, ε∈(0,1) and x_1,…,x_n∈ L_p(μ) with max_i∈{1,…,n} |x_i| _L_p(μ)≤ K, there exists d≤32e^2 (2K)^2plog n/ε^2 and vectors y_1,…, 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 classical factorization theory of L_p(μ) spaces. 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.
READ FULL TEXT