High-Dimensional \(p\)-Norms

Let \(\mathbf {X}=(X_1, \ldots , X_d)\) be a \(\mathbb R^d\)-valued random vector with i.i.d. components, and let \(\Vert \mathbf {X}\Vert _p= (\sum _{j=1}^d|X_j|^p)^{1/p}\) be its \(p\)-norm, for \(p>0\). The impact of letting \(d\) go to infinity on \(\Vert \mathbf {X}\Vert _p\) 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.