2015/11/30 by Kai Zhang
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Combinatorics #Geology #Image and Signal Denoising Methods #Mathematical Approximation and Integration #Mathematics #Rank (graph theory) #Sparse and Compressive Sensing Techniques #cs.IT #math.IT #math.ST #physics.data-an #stat.ME #stat.ML #stat.TH
paper · pdf · doi:10.1109/tit.2017.2700202
14 pages; 1 figure. Accepted Jan 31, 2017 by IEEE Transactions on Information Theory
openalex publication_date 2017/05/02 · arxiv created 2017/05/04 · arxiv updated 2017/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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-mentioned 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.