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.
Another related setting is the study of the critical points of the Euclidean distance function on an algebraic variety; algebraic properties of these critical points are investigated in REF .
The Euclidean distance degree of an algebraic variety
arXiv: Algebraic Geometry
Mathematics Computer Science