Large deviations for high-dimensional random projections of ℓ_pⁿ-balls

2016/08/12 by Alonso-Gutiérrez, David, Prochno, Joscha, Thaele, Christoph
#46B09 #52A23 #60D05 #60F10 #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)

paper · doi:10.48550/arxiv.1608.03863

Abstract

The paper provides a description of the large deviation behavior for the Euclidean norm of projections of ℓ_pⁿ-balls to high-dimensional random subspaces. More precisely, for each integer n≥ 1, let k_n∈\1,…,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 ℓ_pⁿ-ball of \mathbb Rⁿ for some p∈[1,∞]. 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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