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Distance Optimization and the Extremal Variety of the Grassmann Variety

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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.

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Notes

  1. \((\hat{\underline{z}, \underline{x}})\) denotes the angle between multivectors \(\underline{z}\) and \(\underline{x}\).

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Acknowledgments

This article is supported by Marie Curie FP7-PEOPLE-2012-IEF 329084 A-DAP. We would also like to thank the anonymous referees for the valuable and helpful suggestions they made for the improvement of this article and their remarks with respect to the projective and the affine space as well as the determinantal properties of the extremal varieties.

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Authors and Affiliations

  1. University of Athens, Athens, Greece

    John Leventides

  2. City University London, London, UK

    George Petroulakis & Nicos Karcanias

Authors
  1. John Leventides
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  2. George Petroulakis
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  3. Nicos Karcanias
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Correspondence to John Leventides.

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Leventides, J., Petroulakis, G. & Karcanias, N. Distance Optimization and the Extremal Variety of the Grassmann Variety. J Optim Theory Appl 169, 1–16 (2016). https://doi.org/10.1007/s10957-015-0840-7

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  • DOI: https://doi.org/10.1007/s10957-015-0840-7

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