Spherical Cap Packing Asymptotics and Rank-Extreme Detection
This work addresses the problem of detecting low-rank structures in high-dimensional data for statistical analysis, representing an incremental advancement with specific theoretical bounds.
The paper tackles the spherical cap packing problem using a probabilistic approach, deriving asymptotic sharp universal bounds on maximal inner products and applying these to develop a fast detection method for low-rank structures in high-dimensional Gaussian data, achieving results without using spectrum information.
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.