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.

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Journal Foundations of Computational Mathematics
Draisma, J, Horobeţ, E. (Emil), Ottaviani, G. (Giorgio), Sturmfels, B, & Thomas, R.R. (2016). The Euclidean distance degree of an algebraic variety. Foundations of Computational Mathematics, 16(1), 99–149. doi:10.1007/s10208-014-9240-x