Column randomization and almost-isometric embeddings

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