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Large deviations for high-dimensional random projections of $\ell_p^n$-balls (1608.03863v3)

Published 12 Aug 2016 in math.PR and math.FA

Abstract: The paper provides a description of the large deviation behavior for the Euclidean norm of projections of $\ell_pn$-balls to high-dimensional random subspaces. More precisely, for each integer $n\geq 1$, let $k_n\in{1,\ldots,n-1}$, $E{(n)}$ be a uniform random $k_n$-dimensional subspace of $\mathbb Rn$ and $X{(n)}$ be a random point that is uniformly distributed in the $\ell_pn$-ball of $\mathbb Rn$ for some $p\in[1,\infty]$. Then the Euclidean norms $|P_{E{(n)}}X{(n)}|_2$ of the orthogonal projections are shown to satisfy a large deviation principle as the space dimension $n$ tends to infinity. Its speed and rate function are identified, making thereby visible how they depend on $p$ and the growth of the sequence of subspace dimensions $k_n$. As a key tool we prove a probabilistic representation of $|P_{E{(n)}}X{(n)}|_2$ which allows us to separate the influence of the parameter $p$ and the subspace dimension $k_n$.

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