随机矩阵概率不等式类的证明
The Proof of Probability Inequality of Random Matrix
摘要: 在证明随机矩阵满足限制等距性质以及估计限制等距常数时，Khintchine不等式、Hoeffding型不等式以及Bernstein型不等式都发挥着重要的作用。在实际应用当中，这三种不等式根据随机变量类别的不同又有很多变化。本文重点补充证明了当随机变量为矩阵时的Khintchine不等式、Hoeffding型不等式以及Bernstein型不等式，这些不等式在随机测量矩阵稀疏恢复证明过程中起到关键作用。
Abstract: Khintchine’s inequality, Hoeffding’s inequality and Bernstein’s inequality play an important role in proving that random matrix satisfies the property of restricted isometry and estimating the constant of restricted isometry. In practical application, these three inequalities have many changes according to the different types of random variables. In this paper, we mainly prove the Khintchine inequality, Hoeffding type inequality and Bernstein type inequality when the random variable is a matrix. These inequalities play a key role in the proof of sparse restoration of random measurement matrix.
文章引用：郑珂, 宋儒瑛. 随机矩阵概率不等式类的证明[J]. 应用数学进展, 2021, 10(7): 2410-2418. https://doi.org/10.12677/AAM.2021.107253

参考文献

[1] Vershynin, R. (2012) Introduction to the Non-Asymptotic Analysis of Random Matrices. In: Eldar, Y. and Kutyniok, G., Eds., Compressed Sensing: Theory and Applications, Cambridge University Press, Cambridge, xii+544.
[2] Qiu, R. and Wicks, M. (2014) Cognitive Networked Sensing and Big Data. Springer, New York. [Google Scholar] [CrossRef] 
[3] Foucart, S. and Rauhut, H. (2013) A Mathematical Introduction to Compressive Sensing. Springer, New York. [Google Scholar] [CrossRef] 
[4] van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes. Springer, New York. [Google Scholar] [CrossRef] 
[5] Gross, D. (2011) Recovering Low-Rank Matrices from Few Coefficients in Any Basis. IEEE Transactions on Information Theory, 57, 1548-1566. [Google Scholar] [CrossRef] 
[6] Recht, B. (2011) A Simpler Approach to Matrix Completion. The Journal of Machine Learning Research, 12, 3413- 3430.
[7] Oliveira, R. (2009) Concentration of the Adjacency Matrix and of the Laplacian in Random Graphs with Independent Edges. arXiv:0911.0600.
[8] Baraniuk, R., Davenport, M.A., Duarte, M.F. and Hegde, C. (2011) An Introduction to Compressive Sensing. Rice University, Houston.
[9] Rauhut, H. (2010) Compressive Sensing and Structured Random Matrices. In: Fornasier, M., Ed., Theoretical Foundations and Numerical Methods for Sparse Recovery, Radon Series on Computational and Applied Mathematics, de Gruyter, Berlin, Vol. 9, 1-92.
[10] Oliveira, R. (2010) Sums of Random Hermitian Matrices and an Inequality by Rudelson. Electronic Communications in Probability, 15, 203-212. [Google Scholar] [CrossRef] 
[11] Tropp, J.A. (2012) User-Friendly Tail Bounds for Sums of Random Matrices. Foundations of Computational Mathematics, 12, 389-434. [Google Scholar] [CrossRef] 

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