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Abstract:

We use tropical geometry to compute the multidegree and Newton polytope of the hypersurface of a statistical model with two hidden and four observed binary random variables, solving an open question stated by Drton, Sturmfels and Sullivant in (Drton et al., 2009, Ch. VI, Problem 7.7). The model is obtained from the undirected graphical model of the complete bipartite graph K2,4 by marginalizing two of the six binary random variables. We present algorithms for computing the Newton polytope of its defining equation by parallel walks along the polytope and its normal fan. In this way we compute vertices of the polytope. Finally, we also compute and certify its facets by studying tangent cones of the polytope at the symmetry classes of vertices. The Newton polytope has 17. 214. 912 vertices in 44. 938 symmetry classes and 70. 646 facets in 246 symmetry classes. © 2010.

Registro:

Documento: Artículo
Título:An implicitization challenge for binary factor analysis
Autor:Cueto, M.A.; Tobis, E.A.; Yu, J.
Filiación:Department of Mathematics, University of California, Berkeley. 970 Evans Hall #3840, Berkeley, CA 94720-3840, United States
Departamento de Matemática, FCEN - Universidad de Buenos Aires, Pabellón I - Ciudad Universitaria, C1428EGA, Buenos Aires, Argentina
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States
Palabras clave:Factor analysis; Hadamard products; Newton polytope; Tropical geometry
Año:2010
Volumen:45
Número:12
Página de inicio:1296
Página de fin:1315
DOI: http://dx.doi.org/10.1016/j.jsc.2010.06.011
Título revista:Journal of Symbolic Computation
Título revista abreviado:J. Symb. Comput.
ISSN:07477171
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_07477171_v45_n12_p1296_Cueto.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_07477171_v45_n12_p1296_Cueto

Referencias:

  • Bogart, T., Jensen, A.N., Speyer, D., Sturmfels, B., Thomas, R., Computing tropical varieties (2007) J. Symbolic Comput., 42 (1-2), pp. 54-73
  • Cueto, M.A., Morton, J., Sturmfels, B., Geometry of the restricted Boltzmann machine (2010) Contemporary Mathematics, 516, pp. 135-153. , American Mathematical Society, M. Viana, H. Wynn (Eds.) Algebraic Methods in Statistics and Probability
  • De Loera, J.A., Haws, D., Hemmecke, R., Huggins, P., Tauzer, J., Yoshida, R., (2003), http://www.math.ucdavis.edu/~latte, A user's guide for latte v1.1. Available at; Dickenstein, A., (2008), http://www.damtp.cam.ac.uk/user/na/FoCM/FoCM08/Talks/Dickenstein.pdf, A world of binomials. Available at, 2008. Plenary lecture FoCM; Dickenstein, A., Feichtner, E.M., Sturmfels, B., Tropical discriminants (2007) J. Amer. Math. Soc., 20 (4), pp. 1111-1133
  • Drton, M., Sturmfels, B., Sullivant, S., Lectures on Algebraic Statistics (2009) Oberwolfach Seminars, 39. , Birhkäuser
  • Einsiedler, M., Kapranov, M., Lind, D., Non-Archimedean amoebas and tropical varieties (2006) J. Reine Angew. Math., 601, pp. 139-157
  • Eisenbud, D., Commutative algebra with a view toward algebraic geometry (1995) Graduate Texts in Mathematics, 150. , Springer-Verlag, New York
  • Eriksson, N., Ranestad, K., Sturmfels, B., Sullivant, S., Phylogenetic algebraic geometry (2005) Projective Varieties with Unexpected Properties, pp. 237-255. , Walter de Gruyter GmbH & Co, KG, Berlin
  • Galassi, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Alken, P., Booth, M., Rossi, F., (2009), http://www.gnu.org/software/gsl/, GNU Scientific Library Reference Manual - Third Edition. Network Theory Ltd; Gawrilow, E., Joswig, M., Polymake: a framework for analyzing convex polytopes (2000) Polytopes - Combinatorics and Computation, pp. 43-74. , Birkhäuser, G. Kalai, G. Ziegler (Eds.)
  • Grayson, D.R., Stillman, M.E., (2009), http://www.math.uiuc.edu/Macaulay2/, Macaulay2, a software system for research in algebraic geometry. Available at; Jensen, A.N., (2009), http://www.math.tu-berlin.de/~jensen/software/gfan/gfan.html, Gfan, a software system for Gröbner fans and tropical varieties. Available at; Landsberg, J.M., Manivel, L., On the ideals of secant varieties of Segre varieties (2004) Found. Comput. Math., 4 (4), pp. 397-422
  • Landsberg, J.M., Weyman, J., On the ideals and singularities of secant varieties of Segre varieties (2007) Bull. Lond. Math. Soc., 39 (4), pp. 685-697
  • Le Roux, N., Bengio, Y., Representational power of restricted Boltzmann machines and deep belief networks (2008) Neural Comput., 20 (6), pp. 1631-1649
  • Lutz, M., Ascher, D., Willison, F., (1999) Learning Python, , O'Reilly & Associates, Inc, Sebastopol, CA, USA
  • Monagan, M.B., Geddes, K.O., Heal, K.M., Labahn, G., Vorkoetter, S.M., McCarron, J., DeMarco, P., (2005) Maple~10 Programming Guide, , Maplesoft, Waterloo ON, Canada
  • Pachter, L., Sturmfels, B., (2005) Algebraic Statistics for Computational Biology, , Cambridge University Press, New York, NY, USA
  • Richter-Gebert, J., Sturmfels, B., Theobald, T., First steps in tropical geometry (2005) Contemp. Math., 377, pp. 289-317. , Amer. Math. Soc., Providence, RI, Idempotent Mathematics and Mathematical Physics
  • Stroustrup, B., (1997) The C++ Programming Language, , Addison-Wesley, Reading, MA
  • Sturmfels, B., Tevelev, J., Elimination theory for tropical varieties (2008) Math. Res. Lett., 15 (3), pp. 543-562
  • Sturmfels, B., Tevelev, J., Yu, J., The Newton polytope of the implicit equation (2007) Mosc. Math. J., 7 (2), pp. 327-346. , 351
  • Sturmfels, B., Yu, J., Tropical implicitization and mixed fiber polytopes (2008) IMA Vol. Math. Appl., 148, pp. 111-131. , Springer, New York, Software for Algebraic Geometry

Citas:

---------- APA ----------
Cueto, M.A., Tobis, E.A. & Yu, J. (2010) . An implicitization challenge for binary factor analysis. Journal of Symbolic Computation, 45(12), 1296-1315.
http://dx.doi.org/10.1016/j.jsc.2010.06.011
---------- CHICAGO ----------
Cueto, M.A., Tobis, E.A., Yu, J. "An implicitization challenge for binary factor analysis" . Journal of Symbolic Computation 45, no. 12 (2010) : 1296-1315.
http://dx.doi.org/10.1016/j.jsc.2010.06.011
---------- MLA ----------
Cueto, M.A., Tobis, E.A., Yu, J. "An implicitization challenge for binary factor analysis" . Journal of Symbolic Computation, vol. 45, no. 12, 2010, pp. 1296-1315.
http://dx.doi.org/10.1016/j.jsc.2010.06.011
---------- VANCOUVER ----------
Cueto, M.A., Tobis, E.A., Yu, J. An implicitization challenge for binary factor analysis. J. Symb. Comput. 2010;45(12):1296-1315.
http://dx.doi.org/10.1016/j.jsc.2010.06.011