The Euclidean distance degree of an algebraic variety

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.

to appear in Foundations of Computational Mathematics

References in corpus (2)

• Normal class and normal lines of algebraic hypersurfaces
• The average number of critical rank-one approximations to a tensor

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