Abstract
For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator \({S : \ell_p^n \to \mathbb{R}^N}\) such that for every 0 < r < p < 2 with r ≤ 1, the operator \({S_r = S : \ell_p^n \to \ell_r^N}\) satisfies with overwhelming probability that \({\|S_r\| \, \|(S_r)_{| {\rm Im}\, S}^{-1}\| \le C(p,r)^{n/(N-n)}}\), where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.
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Friedland, O., Guédon, O. Random embedding of \({\ell_p^n}\) into \({\ell_r^N}\) . Math. Ann. 350, 953–972 (2011). https://doi.org/10.1007/s00208-010-0581-8
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DOI: https://doi.org/10.1007/s00208-010-0581-8