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.
In REF , the authors show several algebraic and geometric properties of the critical points of the Euclidean distance function on an algebraic variety.
The Euclidean distance degree of an algebraic variety
arXiv: Algebraic Geometry
Mathematics Computer Science