期刊
IEEE TRANSACTIONS ON INFORMATION THEORY
卷 63, 期 7, 页码 4572-4584出版社
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TIT.2017.2700202
关键词
Spherical cap packing; extreme value distribution; spurious correlation; low-rank detection and estimation; high-dimensional inference
资金
- NSF [DMS-1309619, DMS-1613112, IIS-1633212, DMS-1127914]
- Junior Faculty Development Award at UNC Chapel Hill
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1613112] Funding Source: National Science Foundation
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.
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