jmlr jmlr2013 jmlr2013-36 jmlr2013-36-reference knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Tony Cai, Jianqing Fan, Tiefeng Jiang
Abstract: This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in R p as the number of points n → ∞, while the dimension p is either fixed or growing with n. For both settings, we derive the limiting empirical distribution of the random angles and the limiting distributions of the extreme angles. The results reveal interesting differences in the two settings and provide a precise characterization of the folklore that “all high-dimensional random vectors are almost always nearly orthogonal to each other”. Applications to statistics and machine learning and connections with some open problems in physics and mathematics are also discussed. Keywords: random angle, uniform distribution on sphere, empirical law, maximum of random variables, minimum of random variables, extreme-value distribution, packing on sphere
Lars V. Ahlfors. Complex Analysis. McGraw-Hill, New York, 1979. Vangalur S. Alagar. The distribution of the distance between random points. Journal of Applied Probability, 13(3):558–566, 1976. Diego Armentano, Carlos Beltr´ n, and Michael Shub. Minimizing the discrete logarithmic energy a on the sphere: The role of random polynomials. Transactions of the American Mathematical Society, 363(6):2955–2965, 2011. Richard Arratia, Larry Goldstein, and Louis Gordon. Two moments suffice for poisson approximations: the chen-stein method. The Annals of Probability, 17(1):9–25, 1989. Peter J. Bickel, Yaacov Ritov, and Alexandre B. Tsybakov. Simultaneous analysis of lasso and dantzig selector. The Annals of Statistics, 37(4):1705–1732, 2009. 1862 D ISTRIBUTIONS OF A NGLES IN R ANDOM PACKING ON S PHERES Tony T. Cai and Tiefeng Jiang. Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices. The Annals of Statistics, 39(3):1496–1525, 2011. Tony T. Cai and Tiefeng Jiang. Phase transition in limiting distributions of coherence of highdimensional random matrices. Journal of Multivariate Analysis, 107:24–39, 2012. Ery Arias Castro, S´ bastien Bubeck, and G´ bor Lugosi. Detecting positive correlations in a multie a variate sample. arXiv preprint arXiv:1202.5536, 2012. Luc Devroye, Andr´ s Gy¨ rgy, G´ bor Lugosi, and Frederic Udina. High-dimensional random geoa o a metric graphs and their clique number. Electronic Journal of Probability, 16:2481–2508, 2011. Persi Diaconis and David Freedman. Asymptotics of graphical projection pursuit. The Annals of Statistics, 12(3):793–815, 1984. Zhishan Dong, Tiefeng Jiang, and Danning Li. Circular law and arc law for truncation of random unitary matrix. Journal of Mathematical Physics, 53:013301–14, 2012. David L Donoho and Xiaoming Huo. Uncertainty principles and ideal atomic decomposition. IEEE Transactions on Information Theory, 47(7):2845–2862, 2001. Kai-Tai Fang, Samuel Kotz, and Kai Wang Ng. Symmetric Multivariate and Related Distributions. Chapman and Hall Ltd. London., 1990. Theodore W Gamelin. Complex Analysis. Springer New York, 2001. Ricardo Garc´a-Pelayo. Distribution of distance in the spheroid. Journal of Physics A: Mathematical ı and General, 38(16):3475–3482, 2005. Eric Gautier and Alexandre B Tsybakov. High-dimensional instrumental variables regression and confidence sets. arXiv preprint arXiv:1105.2454, 2011. Peter Hall, JS Marron, and Amnon Neeman. Geometric representation of high dimension, low sample size data. Journal of the Royal Statistical Society: Series B, 67(3):427–444, 2005. John M Hammersley. The distribution of distance in a hypersphere. The Annals of Mathematical Statistics, 21(3):447–452, 1950. Tiefeng Jiang. The asymptotic distributions of the largest entries of sample correlation matrices. The Annals of Applied Probability, 14(2):865–880, 2004. Ali Katanforoush and Mehrdad Shahshahani. Distributing points on the sphere, i. Experimental Mathematics, 12(2):199–209, 2003. Wilfrid S Kendall and Ilya Molchanov. New Perspectives in Stochastic Geometry. Oxford University Press, 2010. Arno Kuijlaars and E Saff. Asymptotics for minimal discrete energy on the sphere. Transactions of the American Mathematical Society, 350(2):523–538, 1998. 1863 C AI , FAN AND J IANG Michel Ledoux. The Concentration of Measure Phenomenon. American Mathematical Society, 2001. Deli Li and Andrew Rosalsky. Some strong limit theorems for the largest entries of sample correlation matrices. The Annals of Applied Probability, 16(1):423–447, 2006. Deli Li, Weidong Liu, and Andrew Rosalsky. Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix. Probability Theory and Related Fields, 148(1-2):5–35, 2010. Deli Li, Yongcheng Qi, and Andrew Rosalsky. On jiang’s asymptotic distribution of the largest entry of a sample correlation matrix. Journal of Multivariate Analysis, 111:256–270, 2012. Weidong Liu, Zhengyan Lin, and Qiman Shao. The asymptotic distribution and berry–esseen bound of a new test for independence in high dimension with an application to stochastic optimization. The Annals of Applied Probability, 18(6):2337–2366, 2008. Reginald Douglas Lord. The distribution of distance in a hypersphere. The Annals of Mathematical Statistics, 25(4):794–798, 1954. Robb J Muirhead. Aspects of Multivariate Statistical Theory. Wiley. New York., 1982. Mathew Penrose. Random Geometric Graphs. Oxford University Press. Oxford., 2003. Sidney I Resnick. Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York., 2007. Dan Shen, Haipeng Shen, Hongtu Zhu, and JS Marron. Surprising asymptotic conical structure in critical sample eigen-directions. arXiv preprint arXiv:1303.6171, 2013. Steve Smale. Mathematical problems for the next century. Mathematics: Frontiers and Perspectives (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur), pages 271–294, 2000. Dietrich Stoyan, Wilfrid S Kendall, Joseph Mecke, and DG Kendall. Stochastic Geometry and its Applications (2nd ed.), volume 2. Wiley. Chichester W. Sussex and New York., 1995. Shu-Ju Tu and Ephraim Fischbach. Random distance distribution for spherical objects: general theory and applications to physics. Journal of Physics A: Mathematical and General, 35(31): 6557–6570, 2002. Mike Williams. On a probability distribution function arising in stochastic neutron transport theory. Journal of Physics A: Mathematical and General, 34(22):4653–4662, 2001. Wang Zhou. Asymptotic distribution of the largest off-diagonal entry of correlation matrices. Transactions of the American Mathematical Society, 359(11):5345–5363, 2007. 1864