Abstract
We prove estimates for the expected value of operator norms of Gaussian random matrices with independent (but not necessarily identically distributed) and centered entries, acting as operators from \(\ell_{p^{{\ast}}}^{n}\) to ℓ q m, 1 ≤ p ∗ ≤ 2 ≤ q < ∞.
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References
A. Bandeira, R. van Handel, Sharp nonasymptotic bounds on the norm of random matrices with independent entries. Ann. Probab. 44, 2479–2506 (2016)
G. Bennett, V. Goodman, C.M. Newman, Norms of random matrices. Pac. J. Math. 59 (2), 359–365 (1975)
Y. Benyamini, Y. Gordon, Random factorization of operators between Banach spaces. J. Anal. Math. 39, 45–74 (1981)
S. Chevet, 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), pages Exp. No. 19, 15 (École Polytech., Palaiseau, 1978)
B.S. Cirel’son, I.A. Ibragimov, V.N. Sudakov, Norms of Gaussian sample functions, in Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975). Lecture Notes in Mathematics, vol. 550 (Springer, Berlin, 1976), pp. 20–41
K.R. Davidson, S.J. Szarek, Addenda and corrigenda to: “Local operator theory, random matrices and Banach spaces”, in Handbook of the Geometry of Banach Spaces, vol. 2 (North-Holland, Amsterdam, 2003)
K.R. Davidson, S.J. Szarek, Local operator theory, random matrices and Banach spaces, in Handbook of the Geometry of Banach Spaces, vol. 1 (North-Holland, Amsterdam, 2003)
T. Figiel, On the moduli of convexity and smoothness. Stud. Math. 56 (2), 121–155 (1976)
Y. Gordon, Some inequalities for gaussian processes and applications. Isr. J. Math. 50, 265–289 (1985)
Y. Gordon, A.E. Litvak, C. Schütt, E. Werner, Orlicz norms of sequences of random variables. Ann. Probab. 30 (4), 1833–1853 (2002)
Y. Gordon, A.E. Litvak, C. Schütt, E. Werner, Uniform estimates for order statistics and Orlicz functions. Positivity 16 (1), 1–28 (2012)
O. Guédon, S. Mendelson, A. Pajor, N. Tomczak-Jaegermann, Majorizing measures and proportional subsets of bounded orthonormal systems. Rev. Mat. Iberoam. 24 (3), 1075–1095 (2008)
O. Guédon, M. Rudelson, Moments of random vectors via majorizing measures. Adv. Math. 208 (2), 798–823 (2007)
J. Hoffmann-Jørgensen, Sums of independent Banach space valued random variables. Stud. Math. 52, 159–186 (1974)
R. Latała, Some estimates of norms of random matrices. Proc. Am. Math. Soc. 133 (5), 1273–1282 (electronic) (2005)
M. Ledoux, The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89 (American Mathematical Society, Providence, RI, 2001)
B. Maurey, G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Stud. Math. 58 (1), 45–90 (1976)
G. Pisier, Martingales with values in uniformly convex spaces. Isr. J. Math. 20 (3–4), 326–350 (1975)
G. Pisier, Q. Xu, Non-commutative L p-spaces, in Handbook of the Geometry of Banach Spaces, vol. 2 (North-Holland, Amsterdam, 2003), pp. 1459–1517
S. Riemer, C. Schütt, On the expectation of the norm of random matrices with non-identically distributed entries. Electron. J. Probab. 18 (29), 1–13 (2013)
M. Rudelson, O. Zeitouni, Singular values of gaussian matrices and permanent estimators. Random Struct. Algorithm. 48, 183–212 (2016)
Y. Seginer, The expected norm of random matrices. Combin. Probab. Comput. 9, 149–166 (2000)
J.A. Tropp, User-friendly tail bounds for sums of random matrices. Found. Comput. Math. 12 (4), 389–434 (2012)
R. Van Handel, On the spectral norm of gaussian random matrices. Trans. Am. Math. Soc. (to appear)
J. von Neumann, H.H. Goldstine, Numerical inverting of matrices of high order. Bull. Am. Math. Soc. 53 (11), 1021–1099 (1947)
E.P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62 (3), 548–564 (1955)
E.P. Wigner, On the distribution of the roots of certain symmetric matrices. Ann. Math. 67 (2), 325–327 (1958)
J. Wishart, The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A (1/2), 32–52 (1928)
Acknowledgements
Part of this work was done while Alexander E. Litvak visited Joscha Prochno at the Johannes Kepler University in Linz (supported by FWFM 1628000). Alexander E. Litvak thanks the support of the Bézout Research Foundation (Labex Bézout) for the invitation to the University Marne la Vallée (France).
We would also like to thank our colleague R. Adamczak for helpful comments. We are grateful to the anonymous referee for many useful comments and remarks helping us to improve the presentation as well as for showing the argument outlined in the last section.
J. Prochno was supported in parts by the Austrian Science Fund, FWFM 1628000.
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Guédon, O., Hinrichs, A., Litvak, A.E., Prochno, J. (2017). On the Expectation of Operator Norms of Random Matrices. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2169. Springer, Cham. https://doi.org/10.1007/978-3-319-45282-1_10
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DOI: https://doi.org/10.1007/978-3-319-45282-1_10
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