The Euclidean Distance Degree of an Algebraic Variety

Draisma, Jan; Horobet, Emil; Ottaviani, Giorgio; Sturmfels, Bernd; Thomas, Rekha R.

doi:10.1007/s10208-014-9240-x

Cited by 178 publications

(312 citation statements)

References 38 publications

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“…It is called the generic Euclidean distance degree of X, and denoted by gED(X). We refer to [8] for more details about Euclidean distance degree. For determinantal varieties, we have Proposition 5.5.…”

Section: Proof Of the Main Theorem For Each Coefficientmentioning

confidence: 99%

Chern classes and characteristic cycles of determinantal varieties

Zhang¹

2018

Journal of Algebra

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Abstract. Let K be an algebraically closed field of characteristic 0. For m ≥ n, we define τ m,n,k to be the set of m×n matrices over K with kernel dimension ≥ k. This is a projective subvariety of P mn−1 , and is called the (generic) determinantal variety. In most cases τ m,n,k is singular with singular locus τ m,n,k+1 . In this paper we give explicit formulas computing the Chern-Mather class (c M ) and the Chern-Schwartz-MacPherson class (c SM ) of τ m,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 τ m,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 τ m,n,k τ m,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 τ m,n,k , and the recent computation of the local Euler obstruction of τ m,n,k .

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Section: Proof Of the Main Theorem For Each Coefficientmentioning

confidence: 99%

Chern classes and characteristic cycles of determinantal varieties

Zhang¹

2018

Journal of Algebra

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“…The bulk of this note is devoted to counting the number of critical points on G of the function d u , in the general framework of the Euclidean distance degree (ED degree) [2]. In Section 2 we specialise this framework to matrix groups.…”

Section: The Distance To a Matrix Groupmentioning

confidence: 99%

“…), then the number of solutions to (1) will not depend on u, provided that u is sufficiently general. Following [2], we call this number the Euclidean distance degree (ED degree for short) of G. This number gives an algebraic measure for the complexity of writing down the solution to the minimisation Problem 1.1. We now distinguish two classes of groups: those that preserve the inner product (.|.)…”

Section: The Ed Degree and Critical Equationsmentioning

confidence: 99%

Euclidean distance degrees of real algebraic groups

Baaijens

Draisma

2015

Linear Algebra and its Applications

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“…The following theorem justifies that the ML-degree and ED-degree are well defined. The corresponding proofs can be found in [8] and [7] respectively. Theorem 2.4.…”

Section: Critical Pointsmentioning

confidence: 99%

“…There are already some fundamental results about these algebraic degrees and the geometry behind them. A general degree theory for the ED-degree was introduced in [7] and for the ML-degree in [6,17,19]. For some special cases, it is possible to find formulas for d using techniques from algebraic geometry [1,18,19,21,23,24,25,27], some are summarized in [26].…”

Section: Introductionmentioning

confidence: 99%

Critical points via monodromy and local methods

Campo¹,

Rodriguez²

2017

Journal of Symbolic Computation

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Abstract. In many areas of applied mathematics and statistics, it is a fundamental problem to find the best representative of a model by optimizing an objective function. This can be done by determining critical points of the objective function restricted to the model.We compile ideas arising from numerical algebraic geometry to compute the critical points of an objective function. Our method consists of using numerical homotopy continuation and a monodromy action on the total critical space to compute all of the complex critical points of an objective function. To illustrate the relevance of our method, we apply it to the Euclidean distance function to compute ED-degrees and the likelihood function to compute maximum likelihood degrees.

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The Euclidean Distance Degree of an Algebraic Variety

Cited by 178 publications

References 38 publications

Chern classes and characteristic cycles of determinantal varieties

Chern classes and characteristic cycles of determinantal varieties

Euclidean distance degrees of real algebraic groups

Critical points via monodromy and local methods

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