When fractional quasi p-norms concentrate

TL;DR

This paper investigates the concentration properties of fractional quasi p-norms in high-dimensional data, identifying conditions for concentration and anti-concentration, and clarifying previous controversies.

Contribution

It provides the first comprehensive conditions under which fractional quasi p-norms concentrate or not, resolving longstanding theoretical and empirical debates.

Findings

Fractional quasi p-norms can exhibit exponential and uniform concentration bounds for broad distribution classes.

Certain distributions allow controlling concentration by choosing p appropriately.

Near some distributions, there exist uncountably many others with anti-concentration properties.

Abstract

Concentration of distances in high dimension is an important factor for the development and design of stable and reliable data analysis algorithms. In this paper, we address the fundamental long-standing question about the concentration of distances in high dimension for fractional quasi $p$-norms, $p\in(0,1)$. The topic has been at the centre of various theoretical and empirical controversies. Here we, for the first time, identify conditions when fractional quasi $p$-norms concentrate and when they don't. We show that contrary to some earlier suggestions, for broad classes of distributions, fractional quasi $p$-norms admit exponential and uniform in $p$ concentration bounds. For these distributions, the results effectively rule out previously proposed approaches to alleviate concentration by "optimal" setting the values of $p$ in $(0,1)$. At the same time, we specify conditions and the corresponding families of distributions for which one can still control concentration rates by appropriate choices of $p$. We also show that in an arbitrarily small vicinity of a distribution from a large class of distributions for which uniform concentration occurs, there are uncountably many other distributions featuring anti-concentration properties. Importantly, this behavior enables devising relevant data encoding or representation schemes favouring or discouraging distance concentration. The results shed new light on this long-standing problem and resolve the tension around the topic in both theory and empirical evidence reported in the literature.