The Euclidean Distance Degree of an Algebraic Variety
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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.
KeywordsDistance minimization Computational algebraic geometry Duality Polar classes Low-rank approximation
Mathematics Subject Classification51N35 14N10 14M12 90C26 13P25 15A69
Jan Draisma was supported by a Vidi Grant from the Netherlands Organisation for Scientific Research (NWO), and Emil Horobeţ by the NWO Free Competition Grant Tensors of bounded rank. Giorgio Ottaviani is member of GNSAGA-INDAM. Bernd Sturmfels was supported by the NSF (DMS-0968882), DARPA (HR0011-12-1-0011), and the Max-Planck Institute für Mathematik in Bonn, Germany. Rekha Thomas was supported by the NSF (DMS-1115293).
- 1.C. Aholt, B. Sturmfels and R. Thomas: A Hilbert scheme in computer vision, Canadian J. Mathematics 65 (2013), no. 5, 961–988.Google Scholar
- 3.D. Bates, J. Hauenstein, A. Sommese, and C. Wampler: Numerically Solving Polynomial Systems with Bertini, SIAM, 2013.Google Scholar
- 6.F. Catanese: Caustics of plane curves, their birationality and matrix projections, in Algebraic and Complex Geometry (eds. A. Frühbis-Krüger et al), Springer Proceedings in Mathematics and Statistics 71 (2014) 109–121.Google Scholar
- 9.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.Google Scholar
- 10.J. Draisma and E. Horobeţ: The average number of critical rank-one approximations to a tensor, arxiv:1408.3507.
- 15.S. Friedland and G. Ottaviani: The number of singular vector tuples and uniqueness of best rank one approximation of tensors, Found. Comput. Math., doi: 10.1007/s10208-014-9194-z.
- 17.W. Fulton: Introduction to Toric Varieties, Princeton University Press, 1993.Google Scholar
- 20.D. Grayson and M. Stillman: Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.
- 21.D. Grayson, M. Stillman, S. Strømme, D. Eisenbud, and C. Crissman: Schubert2, computations of characteristic classes for varieties without equations, available at www.math.uiuc.edu/Macaulay2/.
- 23.R. Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977.Google Scholar
- 27.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.Google Scholar
- 28.N.V. Ilyushechkin: The discriminant of the characteristic polynomial of a normal matrix, Mat. Zametki 51 (1992) 16–23; translation in Math. Notes 51(3-4) (1992) 230–235.Google Scholar
- 30.A. Josse and F. Pène: On the normal class of curves and surfaces, arXiv:1402.7266.
- 31.M. Laurent: Cuts, matrix completions and graph rigidity, Mathematical Programming 79 (1997) 255–283.Google Scholar
- 32.L.-H. Lim: Singular values and eigenvalues of tensors: a variational approach, Proc. IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP 05), 1 (2005), 129–132.Google Scholar
- 33.E. Miller and B. Sturmfels: Combinatorial Commutative Algebra, Graduate Texts in Mathematics 227, Springer, New York, 2004.Google Scholar
- 34.The Online Encyclopedia of Integer Sequences, http://oeis.org/.
- 35.G. Ottaviani, P.J. Spaenlehauer, B. Sturmfels: Exact solutions in structured low-rank approximation, SIAM Journal on Matrix Analysis and Applications 35 (2014) 1521–1542.Google Scholar
- 36.P.A. Parrilo: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD Thesis, Caltech, Pasadena, CA, May 2000.Google Scholar
- 38.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.Google Scholar
- 39.G. Salmon: A Treatise on the Higher Plane Curves, Dublin, 1879, available on the web at http://archive.org/details/117724690.
- 41.H. Stewénius, F. Schaffalitzky, and D. Nistér: How hard is 3-view triangulation really?, Proc. International Conference on Computer Vision, Beijing, China (2005) 686–693.Google Scholar
- 42.B. Sturmfels: Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics 97, Amer. Math. Soc., Providence, 2002.Google Scholar
- 44.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, 2005Google Scholar