An application of classical invariant theory to identifiability in nonparametric mixtures

Ryan Elmore[1]; Peter Hall; Amnon Neeman

  • [1] Australian National University, centre for Mathematics and its Applications, Mathematical Sciences Institute, John Dedman Building, Canberra ACT 0200 (AUSTRALIE)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 1, page 1-28
  • ISSN: 0373-0956

Abstract

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It is known that the identifiability of multivariate mixtures reduces to a question in algebraic geometry. We solve the question by studying certain generators in the ring of polynomials in vector variables, invariant under the action of the symmetric group.

How to cite

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Elmore, Ryan, Hall, Peter, and Neeman, Amnon. "An application of classical invariant theory to identifiability in nonparametric mixtures." Annales de l’institut Fourier 55.1 (2005): 1-28. <http://eudml.org/doc/116185>.

@article{Elmore2005,
abstract = {It is known that the identifiability of multivariate mixtures reduces to a question in algebraic geometry. We solve the question by studying certain generators in the ring of polynomials in vector variables, invariant under the action of the symmetric group.},
affiliation = {Australian National University, centre for Mathematics and its Applications, Mathematical Sciences Institute, John Dedman Building, Canberra ACT 0200 (AUSTRALIE)},
author = {Elmore, Ryan, Hall, Peter, Neeman, Amnon},
journal = {Annales de l’institut Fourier},
keywords = {Mixture model; birational; invariant},
language = {eng},
number = {1},
pages = {1-28},
publisher = {Association des Annales de l'Institut Fourier},
title = {An application of classical invariant theory to identifiability in nonparametric mixtures},
url = {http://eudml.org/doc/116185},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Elmore, Ryan
AU - Hall, Peter
AU - Neeman, Amnon
TI - An application of classical invariant theory to identifiability in nonparametric mixtures
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 1
SP - 1
EP - 28
AB - It is known that the identifiability of multivariate mixtures reduces to a question in algebraic geometry. We solve the question by studying certain generators in the ring of polynomials in vector variables, invariant under the action of the symmetric group.
LA - eng
KW - Mixture model; birational; invariant
UR - http://eudml.org/doc/116185
ER -

References

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  1. M. V. Catalisano, A. V. Geramita, A. Gimigliano, Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl. 355 (2002), 263-285 Zbl1059.14061MR1930149
  2. M. V. Catalisano, A. V. Geramita, A. Gimigliano, Erratum to ``Ranks of tensors, secant varieties of Segre varieties and fat points'', Linear Algebra Appl. 367 (2003), 347-348 Zbl1073.14550MR1976931
  3. L. D. Garcia, M. Stillman, B. Sturmfels, Algebraic geometry of Bayesian networks Zbl1126.68102
  4. I. M. Gel'fand, M. M. Kapranov, A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, (1994), Birkhäuser, Boston, MA Zbl0827.14036MR1264417
  5. L. A. Goodman, Exploratory latent structure analysis using both identifiable and unidentifiable models, Biometrika 61 (1974), 215-231 Zbl0281.62057MR370936
  6. P. Hall, X.-H. Zhou, Nonparametric estimation of component distributions in a multivariate mixture, Ann. Statist. 31 (2003), 201-224 Zbl1018.62021MR1962504
  7. P. Hall, A. Neeman, R. Pakyari, R. Elmore, Nonparametric inference in multivariate mixtures Zbl1152.62327
  8. J. M. Landsberg, L. Manivel, On the ideals of secant varieties to Segre varieties Zbl1068.14068
  9. B.G. Lindsay, Mixture Models: Theory Geometry and Applications, (1995), Hayward, Institute of Mathematical Statistics Zbl1163.62326
  10. G.J. Mac, Lachlan, D. Peel, Finite Mixture Models, (2000), John Wiley & Sons 
  11. A. Mattuck, The field of multisymmetric functions, Proc. Amer. Math. Soc. 19 (1968), 764-765 Zbl0159.05303MR225774
  12. M. Nagata, On the normality of the Chow variety of positive 0 -cycles of degree m in an algebraic variety, 29 (1955), 165-176 Zbl0066.14701
  13. A. Neeman, Zero cycles in ℙ n , Advances in Math. 89 (1991), 217-227 Zbl0787.14004MR1128613
  14. E. Netto, Vorlesungen über Algebra, (1896), Leipzig Zbl27.0058.01
  15. H. Teicher, Identifiability of mixtures, Ann. Math. Statist. 32 (1961), 244-248 Zbl0146.39302MR120677
  16. H. Teicher, Identifiability of finite mixtures, Ann. Math. Statist. 34 (1963), 1265-1269 Zbl0137.12704MR155376
  17. D.M. Titterington, A.F. Smith, U.E. Makov, Statistical Analysis of Finite Mixture Distributions, (1985), John Wiley & Sons Zbl0646.62013MR838090
  18. H. Weyl, The Classical Groups. Their Invariants and Representations, (1939), Princeton University Press, Princeton N.J. Zbl0020.20601MR1488158
  19. S.J. Yakowitz, J. D. Spragins, On the identifiability of finite mixtures, Ann. Math. Statist. 39 (1968), 209-214 Zbl0155.25703MR224204

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