TY - JOUR

T1 - The Euclidean distance degree of an algebraic variety

AU - Draisma, J.

AU - Horobet, E.

AU - Ottaviani, G.

AU - Sturmfels, B.

AU - Thomas, R.R.

PY - 2016/2

Y1 - 2016/2

N2 - 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.

AB - 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.

U2 - 10.1007/s10208-014-9240-x

DO - 10.1007/s10208-014-9240-x

M3 - Article

VL - 16

SP - 99

EP - 149

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

IS - 1

ER -