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2017 | OriginalPaper | Buchkapitel

Structured Random Matrices

verfasst von : Ramon van Handel

Erschienen in: Convexity and Concentration

Verlag: Springer New York

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Abstract

Random matrix theory is a well-developed area of probability theory that has numerous connections with other areas of mathematics and its applications. Much of the literature in this area is concerned with matrices that possess many exact or approximate symmetries, such as matrices with i.i.d. entries, for which precise analytic results and limit theorems are available. Much less well understood are matrices that are endowed with an arbitrary structure, such as sparse Wigner matrices or matrices whose entries possess a given variance pattern. The challenge in investigating such structured random matrices is to understand how the given structure of the matrix is reflected in its spectral properties. This chapter reviews a number of recent results, methods, and open problems in this direction, with a particular emphasis on sharp spectral norm inequalities for Gaussian random matrices.

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Vorheriges Kapitel Concentration of Measure Principle and Entropy-Inequalities

Nächstes Kapitel Rates of Convergence for Empirical Spectral Measures: A Soft Approach

The first inequality follows by noting that for every v ∈ B, choosing \(\tilde{v} \in B_{\varepsilon }\) such that \(\|v -\tilde{ v}\| \leq \varepsilon\), we have \(\vert \langle v,Xv\rangle \vert = \vert \langle \tilde{v},X\tilde{v}\rangle +\langle v -\tilde{ v},X(v +\tilde{ v})\rangle \vert \leq \vert \langle \tilde{v},X\tilde{v}\rangle \vert + 2\varepsilon \|X\|\).

For reasons that will become evident in the proof, it is essential to consider (complex) unitary matrices U ₁ ,U ₂ ,U ₃ , despite that all the matrices A _k and X are assumed to be real.

Agarwal, N., Kolla, A., Madan, V.: Small lifts of expander graphs are expanding (2013). Preprint arXiv:1311.3268

Anderson, G.W., Guionnet, A., Zeitouni, O.: An introduction to random matrices, Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2010)MATH

Bai, Z.D., Silverstein, J.W.: No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 (1), 316–345 (1998)MathSciNetCrossRefMATH

Bandeira, A.S., Van Handel, R.: Sharp nonasymptotic bounds on the norm of random matrices with independent entries. Ann. Probab. (2016). To appear

Boucheron, S., Lugosi, G., Massart, P.: Concentration inequalities. Oxford University Press, Oxford (2013)CrossRefMATH

Chevet, S.: Séries de variables aléatoires gaussiennes à valeurs dans \(E\hat{ \otimes }_{\varepsilon }F\). Application aux produits d’espaces de Wiener abstraits. In: Séminaire sur la Géométrie des Espaces de Banach (1977–1978), pp. Exp. No. 19, 15. École Polytech., Palaiseau (1978)

Dirksen, S.: Tail bounds via generic chaining. Electron. J. Probab. 20, no. 53, 29 (2015)

Erdős, L., Yau, H.T.: Universality of local spectral statistics of random matrices. Bull. Amer. Math. Soc. (N.S.) 49 (3), 377–414 (2012)

Gordon, Y.: Some inequalities for Gaussian processes and applications. Israel J. Math. 50 (4), 265–289 (1985)MathSciNetCrossRefMATH

10.

Koltchinskii, V., Lounici, K.: Concentration inequalities and moment bounds for sample covariance operators. Bernoulli (2016). To appear

11.

Latała, R.: Some estimates of norms of random matrices. Proc. Amer. Math. Soc. 133 (5), 1273–1282 (electronic) (2005)

12.

Ledoux, M., Talagrand, M.: Probability in Banach spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 23. Springer-Verlag, Berlin (1991). Isoperimetry and processes

13.

Lust-Piquard, F.: Inégalités de Khintchine dans C _p (1 < p < ∞). C. R. Acad. Sci. Paris Sér. I Math. 303 (7), 289–292 (1986)MathSciNetMATH

14.

Mackey, L., Jordan, M.I., Chen, R.Y., Farrell, B., Tropp, J.A.: Matrix concentration inequalities via the method of exchangeable pairs. Ann. Probab. 42 (3), 906–945 (2014)MathSciNetCrossRefMATH

15.

Marcus, A.W., Spielman, D.A., Srivastava, N.: Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem. Ann. of Math. (2) 182 (1), 327–350 (2015)

16.

Pisier, G.: Probabilistic methods in the geometry of Banach spaces. In: Probability and analysis (Varenna, 1985), Lecture Notes in Math., vol. 1206, pp. 167–241. Springer, Berlin (1986)

17.

Pisier, G.: Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294. Cambridge University Press, Cambridge (2003)CrossRef

18.

Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, New York-London (1975)

19.

Riemer, S., Schütt, C.: On the expectation of the norm of random matrices with non-identically distributed entries. Electron. J. Probab. 18, no. 29, 13 (2013)

20.

Rudelson, M.: Almost orthogonal submatrices of an orthogonal matrix. Israel J. Math. 111, 143–155 (1999)MathSciNetCrossRefMATH

21.

Seginer, Y.: The expected norm of random matrices. Combin. Probab. Comput. 9 (2), 149–166 (2000)MathSciNetCrossRefMATH

22.

Sodin, S.: The spectral edge of some random band matrices. Ann. of Math. (2) 172 (3), 2223–2251 (2010)

23.

Srivastava, N., Vershynin, R.: Covariance estimation for distributions with 2 +ɛ moments. Ann. Probab. 41 (5), 3081–3111 (2013)MathSciNetCrossRefMATH

24.

Talagrand, M.: Upper and lower bounds for stochastic processes, vol. 60. Springer, Heidelberg (2014)CrossRefMATH

25.

Tao, T.: Topics in random matrix theory, Graduate Studies in Mathematics, vol. 132. American Mathematical Society, Providence, RI (2012)CrossRef

26.

Tao, T., Vu, V.: Random matrices: the universality phenomenon for Wigner ensembles. In: Modern aspects of random matrix theory, Proc. Sympos. Appl. Math., vol. 72, pp. 121–172. Amer. Math. Soc., Providence, RI (2014)

27.

Tomczak-Jaegermann, N.: The moduli of smoothness and convexity and the Rademacher averages of trace classes S _p(1 ≤ p < ∞). Studia Math. 50, 163–182 (1974)MathSciNetMATH

28.

Tropp, J.: An introduction to matrix concentration inequalities. Foundations and Trends in Machine Learning (2015)

29.

Tropp, J.: Second-order matrix concentration inequalities (2015). Preprint arXiv:1504.05919

30.

Van Handel, R.: Chaining, interpolation, and convexity. J. Eur. Math. Soc. (2016). To appear

31.

Van Handel, R.: On the spectral norm of Gaussian random matrices. Trans. Amer. Math. Soc. (2016). To appear

32.

Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. In: Compressed sensing, pp. 210–268. Cambridge Univ. Press, Cambridge (2012)

Titel: Structured Random Matrices
verfasst von: Ramon van Handel
Verlag: Springer New York
Buch: Convexity and Concentration
Print ISBN: 978-1-4939-7004-9

Electronic ISBN: 978-1-4939-7005-6

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-1-4939-7005-6_4

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