Large deviations for high-dimensional random projections of l_p^n balls

Alonso Gutiérrez, David (Universidad de Zaragoza) ; Prochno, Joscha ; Thäle, Christoph

Resumen: The paper provides a description of the large deviation behavior for the Euclidean norm of projections of View the MathML sourcelpn-balls to high-dimensional random subspaces. More precisely, for each integer n=1n=1, let kn¿{1,…,n-1}kn¿{1,…,n-1}, E(n)E(n) be a uniform random knkn-dimensional subspace of RnRn and X(n)X(n) be a random point that is uniformly distributed in the View the MathML sourcelpn-ball of RnRn for some p¿[1,8]p¿[1,8]. Then the Euclidean norms ¿PE(n)X(n)¿2¿PE(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 knkn. As a key tool we prove a probabilistic representation of ¿PE(n)X(n)¿2¿PE(n)X(n)¿2 which allows us to separate the influence of the parameter p and the subspace dimension knkn.

Idioma: Inglés

DOI: 10.1016/j.aam.2018.04.003

Año: 2018

Publicado en: ADVANCES IN APPLIED MATHEMATICS 99 (2018), 1-35
ISSN: 0196-8858

Factor impacto JCR: 1.008 (2018)

Categ. JCR: MATHEMATICS, APPLIED rank: 141 / 254 = 0.555 (2018) - Q3 - T2

Factor impacto SCIMAGO: 0.688 - Applied Mathematics (Q2)

Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2016-77710-P

Tipo y forma: Article (PostPrint)

Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)