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The probabilistic estimates on the largest and smallest $ q$-singular values of random matrices


Authors: Ming-Jun Lai and Yang Liu
Journal: Math. Comp. 84 (2015), 1775-1794
MSC (2010): Primary 60B20; Secondary 60F10, 60G50, 60G42
DOI: https://doi.org/10.1090/S0025-5718-2014-02895-0
Published electronically: October 30, 2014
MathSciNet review: 3335891
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Abstract: We study the $ q$-singular values of random matrices with pre-Gaussian entries defined in terms of the $ \ell _{q}$-quasinorm with $ 0<q\le 1$. In this paper, we mainly consider the decay of the lower and upper tail probabilities of the largest $ q$-singular value $ s_{1}^{(q)}$, when the number of rows of the matrices becomes very large. Based on the results in probabilistic estimates on the largest $ q$-singular value, we also give probabilistic estimates on the smallest $ q$-singular value for pre-Gaussian random matrices.


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Additional Information

Ming-Jun Lai
Affiliation: Department of Mathematics, The University of Georgia, Athens, Georgia 30602
Email: mjlai@math.uga.edu

Yang Liu
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 488244-1027
Email: yliu@math.msu.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02895-0
Keywords: Random matrices, probability, pre-Gaussian random variable, generalized singular values
Received by editor(s): November 26, 2012
Received by editor(s) in revised form: September 23, 2013
Published electronically: October 30, 2014
Additional Notes: The first author was partly supported by the National Science Foundation under grant DMS-0713807
The second author was partially supported by the Air Force Office of Scientific Research under grant AFOSR 9550-12-1-0455
Article copyright: © Copyright 2014 American Mathematical Society