Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
A comparison principle for functions of a uniformly random subspace
Download PDF
Download PDF
  • Open Access
  • Published: 07 April 2011

A comparison principle for functions of a uniformly random subspace

  • Joel A. Tropp1 

Probability Theory and Related Fields volume 153, pages 759–769 (2012)Cite this article

  • 519 Accesses

  • 4 Citations

  • Metrics details

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.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. 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)

  2. Chatterjee S., Meckes E.: Multivariate normal approximation using exchangeable pairs. Alea 4, 257–283 (2008)

    MathSciNet  MATH  Google Scholar 

  3. Dvoretsky, A.: Some results on convex bodies and Banach spaces. In: Proc. Intl. Symp. Linear Spaces, pp. 123–160, Jerusalem (1961)

  4. Gordon Y.: Gaussian processes and almost spherical sections of convex bodies. Ann. Probab. 16(1), 180–188 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jiang T.: Maxima of entries of Haar distributed matrices. Probab. Theory Relat. Fields 131, 121–144 (2005)

    Article  MATH  Google Scholar 

  6. Jiang T.: How many entries of a typical orthogonal matrix can be approximated by independent normals?. Ann. Probab. 34(4), 1497–1529 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Johnson W.B., Lindenstrauss J.: Extensions of Lipschitz mappings into a Hilbert space. Contemp. Math. 26, 189–206 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ledoux M., Talagrand M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer, Berlin (1991)

    MATH  Google Scholar 

  9. 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)

    MathSciNet  Google Scholar 

  10. Muirhead R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)

    Book  MATH  Google Scholar 

Download references

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).

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Author information

Authors and Affiliations

  1. California Institute of Technology, Annenberg Center, MC 305-16, Pasadena, CA, 91125-5000, USA

    Joel A. Tropp

Authors
  1. Joel A. Tropp
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joel A. Tropp.

Rights and permissions

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.

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received: 02 February 2011

  • Revised: 24 March 2011

  • Published: 07 April 2011

  • Issue Date: August 2012

  • DOI: https://doi.org/10.1007/s00440-011-0360-9

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (2010)

  • Primary: 60B20
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

Not affiliated

Springer Nature

© 2023 Springer Nature