The Euclidean Distance Degree of an Algebraic Variety

• Published in: Foundations of computational mathematics Vol. 16; no. 1; pp. 99 - 149
• Main Authors: Draisma, Jan, Horobeţ, Emil, Ottaviani, Giorgio, Sturmfels, Bernd, Thomas, Rekha R.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2016; Springer Nature B.V
• Subjects: Algebra; Applications of Mathematics; Computation; Computer Science; Economics; Euclidean geometry; Foundations; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrices (mathematics); Matrix Theory; Numerical Analysis; Singular value decomposition; 14M12; Distance minimization; 14N10; Low-rank approximation; 51N35; Computational algebraic geometry; Duality; 90C26; 13P25; 15A69; Polar classes
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-014-9240-x

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.