# The Euclidean Distance Degree of an Algebraic Variety

```@article{Draisma2016TheED,
title={The Euclidean Distance Degree of an Algebraic Variety},
author={Jan Draisma and Emil Horobet and Giorgio Ottaviani and Bernd Sturmfels and Rekha R. Thomas},
journal={Foundations of Computational Mathematics},
year={2016},
volume={16},
pages={99-149}
}```
• Published 31 August 2013
• Mathematics
• Foundations of Computational Mathematics
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…
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## References

SHOWING 1-10 OF 71 REFERENCES
Likelihood Geometry
• Mathematics
• 2013
We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that
Focal loci of algebraic varieties i
• Mathematics
• 2000
The focal locus ∑x of an affine variety X is roughly speaking the (projective) closure of the set of points O for which there is a smooth point x ∈X and a circle with centre O passing through x which
On the normal class of curves and surfaces
• Mathematics
• 2014
We are interested in the normal class of an algebraic surface S of the complex projective space P^3, that is the number of normal lines to S passing through a generic point of P^3. Thanks to the
Focal Loci of Algebraic Hypersurfaces: A General Theory
AbstractThe focal locus is traditionally defined for a differentiable submanifold of Rn. However, since it depends essentially only on the notion of orthogonality, a focal locus can be also
Normal class and normal lines of algebraic hypersurfaces
• Mathematics
• 2014
We are interested in the normal class of an algebraic hypersurface Z of the complex projective space P^n, that is the number of normal lines to Z passing through a generic point of P^n. Thanks to the
Maximum Likelihood Duality for Determinantal Varieties
• Mathematics
• 2012
In a recent paper, Hauenstein, Sturmfels, and the second author discovered a conjectural bijection between critical points of the likelihood func- tion on the complex variety of matrices of rank r
The geometric and numerical properties of duality in projective algebraic geometry
In this paper we investigate some fundamental geometric and numerical properties ofduality for projective varieties inPkN=PN. We take a point of view which in our opinion is somewhat moregeometric
Varieties with small dual varieties, I
Let X be a complex projective nonlinear n-fold in IP". Let X*_IP m be the dual variety of X. Landman defines the defect of X to be def(X)=N-1- dimX*. For most examples, def(X)=0 (i.e. X* is a
On the Degree of Caustics by Reflection
• Mathematics
• 2012
Given a point S ∈ ℙ2: = ℙ2(ℂ) and an irreducible algebraic curve 𝒞 of ℙ2 (with any type of singularities), we consider the lines ℛ m obtained by reflection of the lines (S m) on 𝒞 (for m ∈ 𝒞). The