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.

翻译：高维空间中距离的集中性是开发和设计稳定可靠数据分析算法的重要因素。本文研究了分数阶拟$p$-范数（$p\in(0,1)$）在高维空间中距离集中性的根本性长期悬而未决问题。该主题一直处于各种理论与实证争议的中心。我们首次给出了分数阶拟$p$-范数何时集中及何时不集中的条件判定。研究证明：与早期部分观点相反，对于广泛分布类，分数阶拟$p$-范数具有指数级且关于$p$一致的集中界。对于这些分布，本文结果实质上排除了此前通过"最优"设置$p\in(0,1)$来缓解集中问题的方案。同时，我们明确了仍可通过适当选择$p$值调控集中率的具体条件及相应分布族。此外，我们证明在满足一致集中条件的大类分布中，任意小的邻域内都存在不可数多个具有反集中特性的其他分布。值得关注的是，这种特性使得设计促进或抑制距离集中性的相关数据编码或表示方案成为可能。本研究为这一长期难题提供了新视角，并调和了文献中理论与实证报告围绕该主题存在的矛盾。