research-article

The euclidean distance degree

Authors:

Giorgio Ottaviani,

Bernd Sturmfels,

Rekha ThomasAuthors Info & Claims

SNC '14: Proceedings of the 2014 Symposium on Symbolic-Numeric Computation

Pages 9 - 16

https://doi.org/10.1145/2631948.2631951

Published: 28 July 2014 Publication History

Abstract

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 computation.

References

[1]

B. Anderson and U. Helmke: Counting critical formations on a line, preprint, 2013.

[2]

D. Bates, J. Hauenstein, A. Sommese, and C. Wampler: Numerically Solving Polynomial Systems with Bertini, SIAM, 2013.

Digital Library

[3]

H.-C.G. von Bothmer and K. Ranestad: A general formula for the algebraic degree in semidefinite programming, Bull. London Math. Soc. 41 (2009) 193--197.

[4]

F. Catanese and C. Trifogli: Focal loci of algebraic varieties I, Communications in Algebra 28 (2000) 6017--6057.

[5]

M. Chu, R. Funderlic, and R. Plemmons: Structured low rank approximation, Linear Algebra Appl. 366 (2003) 157--172.

[6]

D. Cox, J. Little, and D. O'Shea: Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1992.

[7]

M. Drton, B. Sturmfels and S. Sullivant: Lectures on Algebraic Statistics, Oberwolfach Seminars, Vol 39, Birkhäuser, Basel, 2009.

[8]

S. Friedland: Best rank one approximation of real symmetric tensors can be chosen symmetric, Front. Math. China 8(1) (2013) 19--40.

[9]

S. Friedland and G. Ottaviani: The number of singular vector tuples and uniqueness of best rank one approximation of tensors, Foundations of Computational Mathematics, to appear.

[10]

J.-C. Faugère, M. Safey El Din and P.-J. Spaenlehauer: Gröbner bases of bihomogeneous ideals generated by polynomials of bidegree (1, 1): algorithms and complexity, J. Symbolic Comput. 46 (2011) 406--437.

Digital Library

[11]

W. Fulton: Introduction to Toric Varieties, Princeton University Press, 1993.

[12]

D. Grayson and M. Stillman: Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.

[13]

R. Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977.

[14]

D. Hilbert and S. Cohn-Vossen: Anschauliche Geometrie, Springer-Verlag, Berlin, 1932.

[15]

A. Holme: The geometric and numerical properties of duality in projective algebraic geometry, Manuscripta Math. 61 (1988) 145--162.

[16]

S. Hoşten, A. Khetan, and B. Sturmfels: Solving the likelihood equations, Foundations of Computational Mathematics 5 (2005) 389--407.

Digital Library

[17]

J. Huh and B. Sturmfels: Likelihood geometry, in Combinatorial Algebraic Geometry (eds. Aldo Conca et al.), Lecture Notes in Mathematics 2108, Springer, (2014) 63--117.

[18]

E. Miller and B. Sturmfels: Combinatorial Commutative Algebra, Graduate Texts in Mathematics 227, Springer, New York, 2004.

[19]

G. Ottaviani, P-J. Spaenlehauer and B. Sturmfels: Exact solutions in structured low-rank approximation, arXiv: 1311.2376.

[20]

R. Piene: Polar classes of singular varieties, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 247--276.

[21]

P. Rostalski and B. Sturmfels: Dualities, Chapter 5 in G. Blekherman, P. Parrilo and R. Thomas: Semidefinite Optimization and Convex Algebraic Geometry, pp. 203--250, MPS-SIAM Series on Optimization, SIAM, Philadelphia, 2013.

[22]

J. Thomassen, P. Johansen, and T. Dokken: Closest points, moving surfaces, and algebraic geometry, Mathematical methods for curves and surfaces: Tromsø, 2004, 351--362, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2005

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Index Terms

The euclidean distance degree

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Published In

cover image ACM Other conferences

SNC '14: Proceedings of the 2014 Symposium on Symbolic-Numeric Computation

July 2014

154 pages

ISBN:9781450329637

DOI:10.1145/2631948

Editors:
Stephen M. Watt
University of Western Ontario (Canada)
,
Jan Verschelde
University of Illinois at Chicago
,
Lihong Zhi
Chinese Academy of Sciences (China)
,
General Chair:
Lihong Zhi
Chinese Academy of Sciences (China)
,
Program Chair:
Stephen M. Watt
University of Western Ontario (Canada)

Copyright © 2014 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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973 Program: National Basic Research Program of China
KLMM: Key Laboratory of Mathematics Mechanization
MapleSoft
ORCCA: Ontario Research Centre for Computer Algebra
NSFC: Natural Science Foundation of China
Chinese Academy of Engineering: Chinese Academy of Engineering
NAG: Numerical Algorithms Group

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Association for Computing Machinery

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Publication History

Published: 28 July 2014

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SNC '14

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973 Program
KLMM
ORCCA
NSFC
Chinese Academy of Engineering
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SNC '14: Symbolic-Numeric Computation 2014

July 28 - 31, 2014

Shanghai, China

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https://doi.org/10.1007/978-3-031-35924-8_4
Arlé EZizka AKeil PWinter MEssl FKnight TWeigelt PJiménez‐Muñoz MMeyer C(2021) bRacatus : A method to estimate the accuracy and biogeographical status of georeferenced biological data Methods in Ecology and Evolution10.1111/2041-210X.1362912:9(1609-1619)Online publication date: 31-May-2021
https://doi.org/10.1111/2041-210X.13629
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https://doi.org/10.28979/comufbed.597093
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