High-Dimensional $$p$$ p -Norms

Gérard Biau () and David M. Mason ()
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Gérard Biau: Sorbonne Universités, UPMC Univ Paris 06
David M. Mason: University of Delaware

A chapter in Mathematical Statistics and Limit Theorems, 2015, pp 21-40 from Springer

Abstract: Abstract Let $$\mathbf {X}=(X_1, \ldots , X_d)$$ X = ( X 1 , … , X d ) be a $$\mathbb R^d$$ R d -valued random vector with i.i.d. components, and let $$\Vert \mathbf {X}\Vert _p= (\sum _{j=1}^d|X_j|^p)^{1/p}$$ ‖ X ‖ p = ( ∑ j = 1 d | X j | p ) 1 / p be its $$p$$ p -norm, for $$p>0$$ p > 0 . The impact of letting $$d$$ d go to infinity on $$\Vert \mathbf {X}\Vert _p$$ ‖ X ‖ p has surprising consequences, which may dramatically affect high-dimensional data processing. This effect is usually referred to as the distance concentration phenomenon in the computational learning literature. Despite a growing interest in this important question, previous work has essentially characterized the problem in terms of numerical experiments and incomplete mathematical statements. In this paper, we solidify some of the arguments which previously appeared in the literature and offer new insights into the phenomenon.

Keywords: Relative Asymptotic Behavior; Prenorm; Hinneburg; Multimedia Content Retrieval; Neighbor Search Heuristics (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-12442-1_3

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DOI: 10.1007/978-3-319-12442-1_3

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