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 generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.

The euclidean distance degree of an algebraic variety / Jan Draisma; Emil Horobet ̧; Giorgio Ottaviani; Bernd Sturmfels; Rekha R. Thomas. - In: FOUNDATIONS OF COMPUTATIONAL MATHEMATICS. - ISSN 1615-3383. - STAMPA. - 16:(2016), pp. 99-149. [10.1007/s10208-014-9240-x]

The euclidean distance degree of an algebraic variety

OTTAVIANI, GIORGIO MARIA;
2016

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 generic point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.
2016
16
99
149
Jan Draisma; Emil Horobet ̧; Giorgio Ottaviani; Bernd Sturmfels; Rekha R. Thomas
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1038823
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