IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v169y2016i1d10.1007_s10957-015-0840-7.html
   My bibliography  Save this article

Distance Optimization and the Extremal Variety of the Grassmann Variety

Author

Listed:
  • John Leventides

    (University of Athens)

  • George Petroulakis

    (City University London)

  • Nicos Karcanias

    (City University London)

Abstract

The approximation of a multivector by a decomposable one is a distance-optimization problem between the multivector and the Grassmann variety of lines in a projective space. When the multivector diverges from the Grassmann variety, then the approximate solution sought is the worst possible. In this paper, it is shown that the worst solution of this problem is achieved, when the eigenvalues of the matrix representation of a related two-vector are all equal. Then, all these pathological points form a projective variety. We derive the equation describing this projective variety, as well as its maximum distance from the corresponding Grassmann variety. Several geometric and algebraic properties of this extremal variety are examined, providing a new aspect for the Grassmann varieties and the respective projective spaces.

Suggested Citation

  • John Leventides & George Petroulakis & Nicos Karcanias, 2016. "Distance Optimization and the Extremal Variety of the Grassmann Variety," Journal of Optimization Theory and Applications, Springer, vol. 169(1), pages 1-16, April.
    • Handle: RePEc:spr:joptap:v:169:y:2016:i:1:d:10.1007_s10957-015-0840-7
    DOI: 10.1007/s10957-015-0840-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-015-0840-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-015-0840-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

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

    References listed on IDEAS

    as
    1. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer;The Psychometric Society, vol. 1(3), pages 211-218, September.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    Cited by:

    1. J. Leventides & H. Kollias & E. Camouzis & M. Livada, 2021. "Grassmann Inequalities and Extremal Varieties in $${\mathbb {P}}\left( {{ \bigwedge ^p}{\mathbb {R}^n}} \right) $$ P ⋀ p R n," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 836-853, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Sewell, Daniel K., 2018. "Visualizing data through curvilinear representations of matrices," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 255-270.
    2. Kohei Adachi & Nickolay T. Trendafilov, 2016. "Sparse principal component analysis subject to prespecified cardinality of loadings," Computational Statistics, Springer, vol. 31(4), pages 1403-1427, December.
    3. Norman Cliff, 1962. "Analytic rotation to a functional relationship," Psychometrika, Springer;The Psychometric Society, vol. 27(3), pages 283-295, September.
    4. Jushan Bai & Serena Ng, 2020. "Simpler Proofs for Approximate Factor Models of Large Dimensions," Papers 2008.00254, arXiv.org.
    5. Adele Ravagnani & Fabrizio Lillo & Paola Deriu & Piero Mazzarisi & Francesca Medda & Antonio Russo, 2024. "Dimensionality reduction techniques to support insider trading detection," Papers 2403.00707, arXiv.org, revised May 2024.
    6. Alfredo García-Hiernaux & José Casals & Miguel Jerez, 2012. "Estimating the system order by subspace methods," Computational Statistics, Springer, vol. 27(3), pages 411-425, September.
    7. Mitzi Cubilla‐Montilla & Ana‐Belén Nieto‐Librero & Ma Purificación Galindo‐Villardón & Ma Purificación Vicente Galindo & Isabel‐María Garcia‐Sanchez, 2019. "Are cultural values sufficient to improve stakeholder engagement human and labour rights issues?," Corporate Social Responsibility and Environmental Management, John Wiley & Sons, vol. 26(4), pages 938-955, July.
    8. Stegeman, Alwin, 2016. "A new method for simultaneous estimation of the factor model parameters, factor scores, and unique parts," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 189-203.
    9. Jos Berge & Henk Kiers, 1993. "An alternating least squares method for the weighted approximation of a symmetric matrix," Psychometrika, Springer;The Psychometric Society, vol. 58(1), pages 115-118, March.
    10. Shimeng Huang & Henry Wolkowicz, 2018. "Low-rank matrix completion using nuclear norm minimization and facial reduction," Journal of Global Optimization, Springer, vol. 72(1), pages 5-26, September.
    11. Antti J. Tanskanen & Jani Lukkarinen & Kari Vatanen, 2016. "Random selection of factors preserves the correlation structure in a linear factor model to a high degree," Papers 1604.05896, arXiv.org, revised Dec 2018.
    12. Ali Habibnia & Esfandiar Maasoumi, 2021. "Forecasting in Big Data Environments: An Adaptable and Automated Shrinkage Estimation of Neural Networks (AAShNet)," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 363-381, December.
    13. Jin-Xing Liu & Yong Xu & Chun-Hou Zheng & Yi Wang & Jing-Yu Yang, 2012. "Characteristic Gene Selection via Weighting Principal Components by Singular Values," PLOS ONE, Public Library of Science, vol. 7(7), pages 1-10, July.
    14. Kargin, V. & Onatski, A., 2008. "Curve forecasting by functional autoregression," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2508-2526, November.
    15. Yoshio Takane & Forrest Young & Jan Leeuw, 1977. "Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features," Psychometrika, Springer;The Psychometric Society, vol. 42(1), pages 7-67, March.
    16. Aldrin, Magne, 1996. "Moderate projection pursuit regression for multivariate response data," Computational Statistics & Data Analysis, Elsevier, vol. 21(5), pages 501-531, May.
    17. W. Gibson, 1962. "On the least-squares orthogonalization of an oblique transformation," Psychometrika, Springer;The Psychometric Society, vol. 27(2), pages 193-195, June.
    18. Gordana Ispirova & Tome Eftimov & Barbara Koroušić Seljak, 2020. "P-NUT: Predicting NUTrient Content from Short Text Descriptions," Mathematics, MDPI, vol. 8(10), pages 1-21, October.
    19. Chester Harris, 1955. "Separation of data as a principle in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 20(1), pages 23-28, March.
    20. Firoozye, Nikan & Tan, Vincent & Zohren, Stefan, 2023. "Canonical portfolios: Optimal asset and signal combination," Journal of Banking & Finance, Elsevier, vol. 154(C).

    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:spr:joptap:v:169:y:2016:i:1:d:10.1007_s10957-015-0840-7. 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.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.