Euclidean Distance Degree Research Articles

Nearest points on toric varieties

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.

Projective duality and a Chern-Mather involution

We observe that linear relations among Chern-Mather classes of projective varieties are preserved by projective duality. We deduce the existence of an explicit involution on a part of the Chow group of projective space, encoding the effect of duality on Chern-Mather classes. Applications include Plücker formulae, constraints on self-dual varieties, generalizations to singular varieties of classical formulas for the degree of the dual and the dual defect, formulas for the Euclidean distance degree, and computations of Chern-Mather classes and local Euler obstructions for cones.

Chern classes and characteristic cycles of determinantal varieties

Let K be an algebraically closed field of characteristic 0. For m≥n, we define τm,n,k to be the set of m×n matrices over K with kernel dimension ≥k. This is a projective subvariety of Pmn−1, and is called the (generic) determinantal variety. In most cases τm,n,k is singular with singular locus τm,n,k+1. In this paper we give explicit formulas computing the Chern–Mather class (cM) and the Chern–Schwartz–MacPherson class (cSM) of τm,n,k, as classes in the projective space. We also obtain formulas for the conormal cycles and the characteristic cycles of these varieties, and for their generic Euclidean Distance degree. Further, when K=C, we prove that the characteristic cycle of the intersection cohomology sheaf of a determinantal variety agrees with its conormal cycle (and hence is irreducible).Our formulas are based on calculations of degrees of certain Chern classes of the universal bundles over the Grassmannian. For some small values of m,n,k, we use Macaulay2 to exhibit examples of the Chern–Mather classes, the Chern–Schwartz–MacPherson classes and the classes of characteristic cycles of τm,n,k.On the basis of explicit computations in low dimensions, we formulate conjectures concerning the effectivity of the classes and the vanishing of specific terms in the Chern–Schwartz–MacPherson classes of the largest strata τm,n,k∖τm,n,k+1.The irreducibility of the characteristic cycle of the intersection cohomology sheaf follows from the Kashiwara–Dubson's microlocal index theorem, a study of the ‘Tjurina transform’ of τm,n,k, and the recent computation of the local Euler obstruction of τm,n,k.

The euclidean distance degree of orthogonally invariant matrix varieties

The Euclidean distance degree of a real variety is an important invariant arising in distance minimization problems. We show that the Euclidean distance degree of an orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances.

The data singular and the data isotropic loci for affine cones

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.

The Euclidean distance degree of Fermat hypersurfaces

Finding the point in an algebraic variety that is closest to a given point is an optimization problem with many applications. We study the case when the variety is a Fermat hypersurface. Our formula for its Euclidean distance degree is a piecewise polynomial whose pieces are defined by subtle congruence conditions.

The Euclidean Distance Degree of an Algebraic Variety

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.

Counting Real Critical Points of the Distance to Orthogonally Invariant Matrix Sets

Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to compute and count the real smooth critical points of a data matrix on an orthogonally invariant set of matrices. The technique relies on “transfer principles” that allow calculations to be done in the space of singular values of the matrices in the orthogonally invariant set. The calculations often simplify greatly and yield transparent formulas. We illustrate the method on several examples and compare our results to the recently introduced notion of Euclidean distance degree of an algebraic variety.

Euclidean distance degrees of real algebraic groups

We study the problem of finding, in a real algebraic matrix group, the matrix nearest to a given data matrix. We do so from the algebro-geometric perspective of Euclidean distance degrees. We recover several classical results, and among the new results that we prove is a formula for the Euclidean distance degree of special linear groups.

Government approval for fibre winding

We analyze the complexity of fitting a variety, coming from a class of varieties, to a configuration of points in Rn. The complexity measure, called the algebraic complexity, computes the Euclidean Distance Degree (EDdegree) of a certain variety called the hypothesis variety as the number of points in the configuration increases. Finally, we establish a connection to complexity of architectures of polynomial neural networks.For the problem of fitting an (n−1)-sphere to a configuration of m points in Rn, we give a closed formula for the algebraic complexity of the hypothesis variety as m grows for the case of n=1. For the case n>1 we conjecture a generalization of this formula supported by numerical experiments.