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Foundations of Computational Mathematics

, Volume 16, Issue 1, pp 99–149 | Cite as

The Euclidean Distance Degree of an Algebraic Variety

  • Jan Draisma
  • Emil Horobeţ
  • Giorgio Ottaviani
  • Bernd Sturmfels
  • Rekha R. Thomas
Article

Abstract

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low-rank matrices, the Eckart–Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean distance degree of a variety is the number of critical points of the squared distance to a general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.

Keywords

Distance minimization Computational algebraic geometry Duality Polar classes Low-rank approximation 

Mathematics Subject Classification

51N35 14N10 14M12 90C26 13P25 15A69 

Notes

Acknowledgments

Jan Draisma was supported by a Vidi Grant from the Netherlands Organisation for Scientific Research (NWO), and Emil Horobeţ by the NWO Free Competition Grant Tensors of bounded rank. Giorgio Ottaviani is member of GNSAGA-INDAM. Bernd Sturmfels was supported by the NSF (DMS-0968882), DARPA (HR0011-12-1-0011), and the Max-Planck Institute für Mathematik in Bonn, Germany. Rekha Thomas was supported by the NSF (DMS-1115293).

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Copyright information

© SFoCM 2014

Authors and Affiliations

  • Jan Draisma
    • 1
    • 2
  • Emil Horobeţ
    • 1
  • Giorgio Ottaviani
    • 3
  • Bernd Sturmfels
    • 4
  • Rekha R. Thomas
    • 5
  1. 1.TU EindhovenEindhovenThe Netherlands
  2. 2.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands
  3. 3.Università di FirenzeFlorenceItaly
  4. 4.University of CaliforniaBerkeleyUSA
  5. 5.University of WashingtonSeattleUSA

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