The maximum likelihood data singular locus
Pages 99 - 107
Abstract
For general data, the number of complex solutions to the likelihood equations is constant and this number is called the (maximum likelihood) ML-degree of the model. In this article, we describe the special locus of data for which the likelihood equations have a solution in the model's singular locus.
References
[1]
C. Bocci, E. Carlini, J. Kileel, Hadamard products of linear spaces. arXiv:1504.04301
[2]
N. Budur, B. Wang, The signed Euler characteristic of very affine varieties, Int. Math. Res. Not., 14 (2015) 5710-5714.
[3]
F. Catanese, S. Hoşten, A. Khetan, B. Sturmfels, The maximum likelihood degree, Am. J. Math., 128 (2006) 671-697.
[4]
M.A. Cueto, J. Morton, B. Sturmfels, Geometry of the restricted Boltzmann machine, in: Contemp. Math., vol. 516, Amer. Math. Soc., Providence, RI, 2010, pp. 135-153.
[5]
M.A. Cueto, E.A. Tobis, J. Yu, An implicitization challenge for binary factor analysis, J. Symb. Comput., 45 (2010) 1296-1315.
[6]
J. Draisma, E. Horobet, G. Ottaviani, B. Sturmfels, R.R. Thomas, The Euclidean distance degree of an algebraic variety, Found. Comput. Math. (2014).
[7]
D.R. Grayson, M.E. Stillman, Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
[8]
E. Gross, J.I. Rodriguez, Maximum likelihood geometry in the presence of data zeros, in: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ACM, 2014, pp. 232-239.
[9]
J. Hauenstein, J.I. Rodriguez, B. Sturmfels, Maximum likelihood for matrices with rank constraints, J. Algebraic Stat., 5 (2014) 18-38.
[10]
E. Horobe¿, The data singular and the data isotropic loci for affine cones. arXiv:1507.02923
[11]
S. Hoşten, A. Khetan, B. Sturmfels, Solving the likelihood equations, Found. Comput. Math., 5 (2005) 389-407.
[12]
J. Huh, The maximum likelihood degree of a very affine variety, Compos. Math., 149 (2013) 1245-1266.
[13]
J. Huh, Varieties with maximum likelihood degree one, J. Algebraic Stat., 5 (2014) 1-17.
[14]
J. Huh, B. Sturmfels, Likelihood geometry, in: Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 63-117.
[15]
J.I. Rodriguez, Maximum likelihood for dual varieties, in: Proceedings of the 2014 Symposium on Symbolic-Numeric Computation, ACM, 2014, pp. 43-49.
[16]
J.I. Rodriguez, X. Tang, Data-discriminants of likelihood equations, in: Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation, ACM, 2015, pp. 307-314.
[17]
J.I. Rodriguez, B. Wang, The maximum likelihood degree of rank 2 matrices via Euler characteristics. arXiv:1505.06536
[18]
P. Rostalski, B. Sturmfels, Dualities, in: MOS-SIAM Ser. Optim., vol. 13, SIAM, Philadelphia, PA, 2013, pp. 203-249.
[19]
B. Wang, Maximum likelihood degree of Fermat hypersurfaces via Euler characteristics. arXiv:1509.03762
- The maximum likelihood data singular locus
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Published: 01 March 2017
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