Matrix deviation inequality for ℓp-norm

Abstract

Motivated by the general matrix deviation inequality for i.i.d. ensemble Gaussian matrix [R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2018), doi:10.1017/9781108231596 of Theorem 11.1.5], we show that this property holds for the ${\ell }_p$-norm with $1\le p<\infty$ and i.i.d. ensemble sub-Gaussian matrices, i.e. random matrices with i.i.d. mean-zero, unit variance, sub-Gaussian entries. As a consequence of our result, we establish the Johnson–Lindenstrauss lemma from ${\ell }_2^n$-space to ${\ell }_p^m$-space for all i.i.d. ensemble sub-Gaussian matrices.