# Column randomization and almost-isometric embeddings

The matrix A:ℝ^n →ℝ^m is (δ,k)-regular if for any k-sparse vector x, | Ax_2^2-x_2^2| ≤δ√(k)x_2^2. We show that if A is (δ,k)-regular for 1 ≤ k ≤ 1/δ^2, then by multiplying the columns of A by independent random signs, the resulting random ensemble A_ϵ acts on an arbitrary subset T ⊂ℝ^n (almost) as if it were gaussian, and with the optimal probability estimate: if ℓ_*(T) is the gaussian mean-width of T and d_T=sup_t ∈ Tt_2, then with probability at least 1-2exp(-c(ℓ_*(T)/d_T)^2), sup_t ∈ T| A_ϵ t_2^2-t_2^2 | ≤ C(Λ d_T δℓ_*(T)+(δℓ_*(T))^2 ), where Λ=max{1,δ^2log(nδ^2)}. This estimate is optimal for 0<δ≤ 1/√(log n).

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