Geometric sharp large deviations for random projections of ℓ_pⁿ spheres and balls

2020/01/13 by Yin-Ting Liao, Liao, Yin-Ting, Kavita Ramanan +1 · 1 citation
Computer Science · Mathematics · #41A60 (Secondary) #52A23 (Primary) 46B06 #60F10 #FOS: Mathematics #Functional Analysis (math.FA) #Geometry and complex manifolds #Metric Geometry (math.MG) #Point processes and geometric inequalities #Probability (math.PR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2001.04053

openalex publication_date 2020/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. Whereas large deviation principles identify the asymptotic exponential decay rate of probabilities, sharp large deviation estimates also provide the "prefactor" in front of the exponentially decaying term. For fixed p ∈ (1,∞), consider independent sequences (X^(n,p))_{n ∈ ℕ} and (Θⁿ)_{n ∈ ℕ} of random vectors with Θⁿ distributed according to the normalized cone measure on the unit ℓ₂ⁿ sphere, and X^(n,p) distributed according to the normalized cone measure on the unit ℓ_pⁿ sphere. For almost every realization (θⁿ)_n∈ℕ of (Θⁿ)_n∈ℕ, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of X^(n,p) onto θⁿ, that are asymptotically exact (as the dimension n tends to infinity). Furthermore, the case when (X^(n,p))_{n ∈ ℕ} is replaced with (\mathscrX^(n,p))_{n ∈ ℕ}, where \mathscrX^(n,p) is distributed according to the uniform (or normalized volume) measure on the unit ℓ_pⁿ ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions (θⁿ)_{n ∈ℕ} that encodes additional geometric information that enables one to distinguish between projections of balls and spheres. Moreover, comparison with numerical estimates obtained by direct computation and importance sampling shows that the obtained analytical expressions for tail probabilities provide good approximations even for moderate values of n.

Citations

• A conditional strong large deviation result and a functional central limit theorem for the rate function

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