Abstract
ABSTRACTThe generic number of critical points of the Euclidean distance function from a data point to a variety is called the Euclidean distance degree (or ED degree). The two special loci of the data points where the number of critical points is smaller than the ED degree are called the Euclidean distance data singular locus and the Euclidean distance data isotropic locus. In this article, we present connections between these two special loci of an affine cone and its dual cone.
Highlights
Models in science are often expressed as real solution sets of systems of polynomial equations, namely real algebraic varieties
One of the most fundamental optimization problems that can be formulated on such sets is the following: given a real algebraic variety and given a general data point of the ambient space, minimize the Euclidean distance from the given data point to the variety
In order to solve this problem algebraically we examine the critical points of the squared Euclidean distance function
Summary
Models in science are often expressed as real solution sets of systems of polynomial equations, namely real algebraic varieties. In order to solve this problem algebraically we examine the critical points of the squared Euclidean distance function The number of such critical points expresses the algebraic degree of the complexity of writing the optimal solution to the distance minimization problem and it is called the Euclidean Distance Degree. If one of the critical points has a multiplicity, the number of real critical points typically changes, this locus is called the ED-discriminant (or classically focal loci ) and it was studied in [2, 3, 5, 8, 12]. In this article we want to discuss the locus (different from the ED discriminant) of exceptional data points u for which the number of complex critical points is smaller the ED degree. In this article we aim to describe the data singular and the data isotropic loci of affine cones
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