Column randomization and almost-isometric embeddings

The matrix $A:\mathbb{R}^n \to \mathbb{R}^m$ is $(δ,k)$-regular if for any $k$-sparse vector $x$, $\left| \|Ax\|_2^2-\|x\|_2^2\right| \leq δ\sqrt{k} \|x\|_2^2.$ We show that if $A$ is $(δ,k)$-regular for $1 \leq k \leq 1/δ^2$, then by multiplying the columns of $A$ by independent random signs, the resulting random ensemble $A_ε$ acts on an arbitrary subset $T \subset \mathbb{R}^n$ (almost) as if it were gaussian, and with the optimal probability estimate: if $\ell_*(T)$ is the gaussian mean-width of $T$ and $d_T=\sup_{t \in T} \|t\|_2$, then with probability at least $1-2\exp(-c(\ell_*(T)/d_T)^2)$, $\sup_{t \in T} \left| \|A_εt\|_2^2-\|t\|_2^2 \right| \leq C\left(Λd_T δ\ell_*(T)+(δ\ell_*(T))^2 \right),$ where $Λ=\max\{1,δ^2\log(nδ^2)\}$. This estimate is optimal for $0<δ\leq 1/\sqrt{\log n}$.

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