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- W4210334818 abstract "Abstract The matrix $A:{{mathbb{R}}}^n to{{mathbb{R}}}^m$ is $(delta ,k)$-regular if for any $k$-sparse vector $x$, $$begin{align*} & left| |Ax|_2^2-|x|_2^2right| leq delta sqrt{k} |x|_2^2. end{align*}$$We show that if $A$ is $(delta ,k)$-regular for $1 leq k leq 1/delta ^2$, then by multiplying the columns of $A$ by independent random signs, the resulting random ensemble $A_varepsilon $ 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 $1-2exp (-c(ell _*(T)/d_T)^2)$, $$begin{align*} & sup_{t in T} left| |A_varepsilon t|_2^2-|t|_2^2 right| leq Cleft(varLambda d_T deltaell_*(T)+(delta ell_*(T))^2 right), end{align*}$$where $varLambda =max {1,delta ^2log (ndelta ^2)}$. This estimate is optimal for $0<delta leq 1/sqrt{log n}$." @default.
- W4210334818 created "2022-02-08" @default.
- W4210334818 creator A5053455526 @default.
- W4210334818 date "2022-02-02" @default.
- W4210334818 modified "2023-09-25" @default.
- W4210334818 title "Column randomization and almost-isometric embeddings" @default.
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- W4210334818 doi "https://doi.org/10.1093/imaiai/iaab028" @default.
- W4210334818 hasPublicationYear "2022" @default.
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