Spherical Cap Packing Asymptotics and Rank-Extreme Detection

TL;DR

This paper investigates the asymptotic behavior of spherical cap packing and maximal inner products, providing universal bounds and distribution limits, with applications to correlation and low-rank structure detection in high-dimensional data.

Contribution

It introduces new asymptotic bounds and distributional results for spherical cap packing and maximal inner products, enabling fast low-rank detection methods in high-dimensional settings.

Findings

Universal asymptotic bounds on maximal inner products

Distributional limits for extreme values of inner products

A fast method for low-rank structure detection

Abstract

We study the spherical cap packing problem with a probabilistic approach. Such probabilistic considerations result in an asymptotic sharp universal uniform bound on the maximal inner product between any set of unit vectors and a stochastically independent uniformly distributed unit vector. When the set of unit vectors are themselves independently uniformly distributed, we further develop the extreme value distribution limit of the maximal inner product, which characterizes its uncertainty around the bound. As applications of the above asymptotic results, we derive (1) an asymptotic sharp universal uniform bound on the maximal spurious correlation, as well as its uniform convergence in distribution when the explanatory variables are independently Gaussian distributed; and (2) an asymptotic sharp universal bound on the maximum norm of a low-rank elliptically distributed vector, as well as related limiting distributions. With these results, we develop a fast detection method for a low-rank structure in high-dimensional Gaussian data without using the spectrum information.