IDEAS home Printed from https://ideas.repec.org/a/oup/biomet/v94y2007i3p760-766.html
   My bibliography  Save this article

The high-dimension, low-sample-size geometric representation holds under mild conditions

Author

Listed:
  • Jeongyoun Ahn
  • J. S. Marron
  • Keith M. Muller
  • Yueh-Yun Chi

Abstract

High-dimension, low-small-sample size datasets have different geometrical properties from those of traditional low-dimensional data. In their asymptotic study regarding increasing dimensionality with a fixed sample size, Hall et al. (2005) showed that each data vector is approximately located on the vertices of a regular simplex in a high-dimensional space. A perhaps unappealing aspect of their result is the underlying assumption which requires the variables, viewed as a time series, to be almost independent. We establish an equivalent geometric representation under much milder conditions using asymptotic properties of sample covariance matrices. We discuss implications of the results, such as the use of principal component analysis in a high-dimensional space, extension to the case of nonindependent samples and also the binary classification problem. Copyright 2007, Oxford University Press.

Suggested Citation

  • Jeongyoun Ahn & J. S. Marron & Keith M. Muller & Yueh-Yun Chi, 2007. "The high-dimension, low-sample-size geometric representation holds under mild conditions," Biometrika, Biometrika Trust, vol. 94(3), pages 760-766.
    • Handle: RePEc:oup:biomet:v:94:y:2007:i:3:p:760-766
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1093/biomet/asm050
    Download Restriction: Access to full text is restricted to subscribers.
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Chung, Hee Cheol & Ahn, Jeongyoun, 2021. "Subspace rotations for high-dimensional outlier detection," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    2. Yata, Kazuyoshi & Aoshima, Makoto, 2012. "Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 193-215.
    3. Wang, Shao-Hsuan & Huang, Su-Yun & Chen, Ting-Li, 2020. "On asymptotic normality of cross data matrix-based PCA in high dimension low sample size," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    4. Makoto Aoshima & Kazuyoshi Yata, 2014. "A distance-based, misclassification rate adjusted classifier for multiclass, high-dimensional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(5), pages 983-1010, October.
    5. Yata, Kazuyoshi & Aoshima, Makoto, 2013. "PCA consistency for the power spiked model in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 334-354.
    6. Jung, Sungkyu & Sen, Arusharka & Marron, J.S., 2012. "Boundary behavior in High Dimension, Low Sample Size asymptotics of PCA," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 190-203.
    7. Wang, Shao-Hsuan & Huang, Su-Yun, 2022. "Perturbation theory for cross data matrix-based PCA," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    8. Zhao, Junguang & Xu, Xingzhong, 2016. "A generalized likelihood ratio test for normal mean when p is greater than n," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 91-104.
    9. Kazuyoshi Yata & Makoto Aoshima, 2012. "Inference on High-Dimensional Mean Vectors with Fewer Observations Than the Dimension," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 459-476, September.
    10. Jung, Sungkyu, 2018. "Continuum directions for supervised dimension reduction," Computational Statistics & Data Analysis, Elsevier, vol. 125(C), pages 27-43.
    11. Modarres, Reza, 2022. "A high dimensional dissimilarity measure," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    12. Jun Li, 2018. "Asymptotic normality of interpoint distances for high-dimensional data with applications to the two-sample problem," Biometrika, Biometrika Trust, vol. 105(3), pages 529-546.
    13. Yata, Kazuyoshi & Aoshima, Makoto, 2013. "Correlation tests for high-dimensional data using extended cross-data-matrix methodology," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 313-331.
    14. Nakayama, Yugo & Yata, Kazuyoshi & Aoshima, Makoto, 2021. "Clustering by principal component analysis with Gaussian kernel in high-dimension, low-sample-size settings," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    15. Yugo Nakayama & Kazuyoshi Yata & Makoto Aoshima, 2020. "Bias-corrected support vector machine with Gaussian kernel in high-dimension, low-sample-size settings," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(5), pages 1257-1286, October.
    16. Miecznikowski Jeffrey C. & Gaile Daniel P. & Chen Xiwei & Tritchler David L., 2016. "Identification of consistent functional genetic modules," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 15(1), pages 1-18, March.
    17. Shen, Dan & Shen, Haipeng & Marron, J.S., 2013. "Consistency of sparse PCA in High Dimension, Low Sample Size contexts," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 317-333.
    18. Yata, Kazuyoshi & Aoshima, Makoto, 2010. "Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2060-2077, October.
    19. Lee, Myung Hee, 2012. "On the border of extreme and mild spiked models in the HDLSS framework," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 162-168.
    20. Borysov, Petro & Hannig, Jan & Marron, J.S., 2014. "Asymptotics of hierarchical clustering for growing dimension," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 465-479.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:oup:biomet:v:94:y:2007:i:3:p:760-766. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Oxford University Press (email available below). General contact details of provider: https://academic.oup.com/biomet .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.