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