Abstract
This note demonstrates that it is possible to bound the expectation of an arbitrary norm of a random matrix drawn from the Stiefel manifold in terms of the expected norm of a standard Gaussian matrix with the same dimensions. A related comparison holds for any convex function of a random matrix drawn from the Stiefel manifold. For certain norms, a reversed inequality is also valid.
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References
Ball, K.: An elementary introduction to modern convex geometry. In: Flavors of Geometry. Math. Sci. Res. Inst. Publ., vol. 31, pp. 1–58. Cambridge University Press (1997)
Chatterjee S., Meckes E.: Multivariate normal approximation using exchangeable pairs. Alea 4, 257–283 (2008)
Dvoretsky, A.: Some results on convex bodies and Banach spaces. In: Proc. Intl. Symp. Linear Spaces, pp. 123–160, Jerusalem (1961)
Gordon Y.: Gaussian processes and almost spherical sections of convex bodies. Ann. Probab. 16(1), 180–188 (1988)
Jiang T.: Maxima of entries of Haar distributed matrices. Probab. Theory Relat. Fields 131, 121–144 (2005)
Jiang T.: How many entries of a typical orthogonal matrix can be approximated by independent normals?. Ann. Probab. 34(4), 1497–1529 (2006)
Johnson W.B., Lindenstrauss J.: Extensions of Lipschitz mappings into a Hilbert space. Contemp. Math. 26, 189–206 (1984)
Ledoux M., Talagrand M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin (1991)
Milman V.D.: A new proof of A. Dvoretsky’s theorem on cross-sections of convex bodies. Funkcional. Anal. i Priložen 5(4), 28–37 (1971)
Muirhead R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)
Acknowledgments
The author would like to thank Ben Recht and Michael Todd for encouraging him to refine and present these results. Alex Gittens and Tiefeng Jiang provided useful comments on a preliminary draft of this article. The anonymous referees offered several valuable comments. This work has been supported in part by ONR awards N00014-08-1-0883 and N00014-11-1-0025, AFOSR award FA9550-09-1-0643, and a Sloan Fellowship. Some of the research took place at Banff International Research Station (BIRS).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Tropp, J.A. A comparison principle for functions of a uniformly random subspace. Probab. Theory Relat. Fields 153, 759–769 (2012). https://doi.org/10.1007/s00440-011-0360-9
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DOI: https://doi.org/10.1007/s00440-011-0360-9