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 general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.
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C. Aholt, B. Sturmfels and R. Thomas: A Hilbert scheme in computer vision, Canadian J. Mathematics 65 (2013), no. 5, 961–988.
B. Anderson and U. Helmke: Counting critical formations on a line, SIAM J. Control Optim. 52 (2014) 219–242.
D. Bates, J. Hauenstein, A. Sommese, and C. Wampler: Numerically Solving Polynomial Systems with Bertini, SIAM, 2013.
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.
D. Cartwright and B. Sturmfels: The number of eigenvectors of a tensor, Linear Algebra and its Applications 438 (2013) 942–952.
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.
F. Catanese and C. Trifogli: Focal loci of algebraic varieties I, Commun. Algebra 28 (2000) 6017–6057.
M. Chu, R. Funderlic, and R. Plemmons: Structured low rank approximation, Linear Algebra Appl. 366 (2003) 157–172.
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.
J. Draisma and E. Horobeţ: The average number of critical rank-one approximations to a tensor, arxiv:1408.3507.
J. Draisma and J. Rodriguez: Maximum likelihood duality for determinantal varieties, Int. Math. Res. Not. 20 (2014) 5648–5666.
M. Drton, B. Sturmfels and S. Sullivant: Lectures on Algebraic Statistics, Oberwolfach Seminars, Vol 39, Birkhäuser, Basel, 2009.
L. Ein: Varieties with small dual varieties, I, Invent. Math. 86(1) (1986) 63–74.
S. Friedland: Best rank one approximation of real symmetric tensors can be chosen symmetric, Front. Math. China 8(1) (2013) 19–40.
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.
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.
W. Fulton: Introduction to Toric Varieties, Princeton University Press, 1993.
W. Fulton: Intersection Theory, Springer, Berlin, 1998.
I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky: Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994.
D. Grayson and M. Stillman: Macaulay2, a software system for research in algebraic geometry, available at www.math.uiuc.edu/Macaulay2/.
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/.
R. Hartley and P. Sturm: Triangulation, Computer Vision and Image Understanding: CIUV (1997) 68(2): 146–157.
R. Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York, 1977.
D. Hilbert and S. Cohn-Vossen: Anschauliche Geometrie, Springer-Verlag, Berlin, 1932.
A. Holme: The geometric and numerical properties of duality in projective algebraic geometry, Manuscripta Math. 61 (1988) 145–162.
S. Hoşten, A. Khetan, and B. Sturmfels: Solving the likelihood equations, Foundations of Computational Mathematics 5 (2005) 389–407.
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.
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.
A. Josse and F. Pène: On the degree of caustics by reflection, Commun. Algebra 42 (2014), 2442–2475.
A. Josse and F. Pène: On the normal class of curves and surfaces, arXiv:1402.7266.
M. Laurent: Cuts, matrix completions and graph rigidity, Mathematical Programming 79 (1997) 255–283.
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.
E. Miller and B. Sturmfels: Combinatorial Commutative Algebra, Graduate Texts in Mathematics 227, Springer, New York, 2004.
The Online Encyclopedia of Integer Sequences, http://oeis.org/.
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.
P.A. Parrilo: Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization, PhD Thesis, Caltech, Pasadena, CA, May 2000.
R. Piene: Polar classes of singular varieties, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 247–276.
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.
G. Salmon: A Treatise on the Higher Plane Curves, Dublin, 1879, available on the web at http://archive.org/details/117724690.
A. Stegeman and P. Comon: Subtracting a best rank-1 approximation does not necessarily decrease tensor rank, Linear Algebra Appl. 433 (2010) 1276–1300.
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.
B. Sturmfels: Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics 97, Amer. Math. Soc., Providence, 2002.
T. Tao and V. Vu: A central limit theorem for the determinant of a Wigner matrix, Adv. Math. 231 (2012) 74–101.
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
C. Trifogli: Focal loci of algebraic hypersurfaces: a general theory, Geometriae Dedicata 70 (1998) 1–26.
J. Weyman: Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Mathematics, 14, Cambridge University Press, Cambridge, 2003.
Acknowledgments
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).
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Communicated by James Renegar.
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Draisma, J., Horobeţ, E., Ottaviani, G. et al. The Euclidean Distance Degree of an Algebraic Variety. Found Comput Math 16, 99–149 (2016). https://doi.org/10.1007/s10208-014-9240-x
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DOI: https://doi.org/10.1007/s10208-014-9240-x