An application of classical invariant theory to identifiability in nonparametric mixtures
Annales de l'Institut Fourier, Volume 55 (2005) no. 1, p. 1-28

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.

On sait que l'identifiabilité des mélanges multivariés se réduit à une question de géométrie algébrique. Nous résolvons cette question en étudiant des générateurs particuliers dans l'anneau des polynômes à variables vectorielles, invariants sous l'action du groupe symétrique.

DOI : https://doi.org/10.5802/aif.2087
Classification:  13A50,  62G07,  62H12
Keywords: Mixture model, birational, invariant
@article{AIF_2005__55_1_1_0,
     author = {Elmore, Ryan and Hall, Peter and Neeman, Amnon},
     title = {An application of classical invariant theory to identifiability in nonparametric mixtures},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {1},
     year = {2005},
     pages = {1-28},
     doi = {10.5802/aif.2087},
     zbl = {02162462},
     mrnumber = {2141286},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2005__55_1_1_0}
}
Elmore, Ryan; Hall, Peter; Neeman, Amnon. An application of classical invariant theory to identifiability in nonparametric mixtures. Annales de l'Institut Fourier, Volume 55 (2005) no. 1, pp. 1-28. doi : 10.5802/aif.2087. http://www.numdam.org/item/AIF_2005__55_1_1_0/

[1] M. V. Catalisano; A. V. Geramita; A. Gimigliano Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl., Tome 355 (2002), pp. 263-285 | Article | MR 1930149 | Zbl 1059.14061

[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., Tome 367 (2003), p. 347-348 | Article | MR 1976931 | Zbl 01917824

[3] L. D. Garcia; M. Stillman; B. Sturmfels Algebraic geometry of Bayesian networks (e-print, http://arXiv.org/abs/math.AG/0301255)

[4] I. M. Gel'Fand; M. M. Kapranov; A. V. Zelevinsky Discriminants, resultants, and multidimensional determinants, Birkhäuser, Boston, MA, Mathematics: Theory \& Applications (1994) | MR 1264417 | Zbl 0827.14036

[5] L. A. Goodman Exploratory latent structure analysis using both identifiable and unidentifiable models, Biometrika, Tome 61 (1974), pp. 215-231 | Article | MR 370936 | Zbl 0281.62057

[6] P. Hall; X.-H. Zhou Nonparametric estimation of component distributions in a multivariate mixture, Ann. Statist., Tome 31 (2003), pp. 201-224 | Article | MR 1962504 | Zbl 1018.62021

[7] P. Hall; A. Neeman; R. Pakyari; R. Elmore Nonparametric inference in multivariate mixtures (To appear)

[8] J. M. Landsberg; L. Manivel On the ideals of secant varieties to Segre varieties (e-print, http://arXiv.org/abs/math.AG/0311388) | Zbl 1068.14068

[9] B.G. Lindsay Mixture Models: Theory Geometry and Applications, Hayward, Institute of Mathematical Statistics (1995) | Zbl 1163.62326

[10] G.J. Mac; Lachlan; D. Peel Finite Mixture Models, John Wiley & Sons (2000)

[11] A. Mattuck The field of multisymmetric functions, Proc. Amer. Math. Soc., Tome 19 (1968), p. 764-765 | MR 225774 | Zbl 0159.05303

[12] M. Nagata On the normality of the Chow variety of positive 0-cycles of degree m in an algebraic variety (Mem. Coll. Sci. Univ. Kyoto A. Math.) Tome 29 (1955), pp. 165-176 | Zbl 0066.14701

[13] A. Neeman Zero cycles in n , Advances in Math., Tome 89 (1991), pp. 217-227 | Article | MR 1128613 | Zbl 0787.14004

[14] E. Netto Vorlesungen über Algebra, Leipzig, Teubner Verlag (1896) | JFM 27.0058.01

[15] H. Teicher Identifiability of mixtures, Ann. Math. Statist., Tome 32 (1961), pp. 244-248 | Article | MR 120677 | Zbl 0146.39302

[16] H. Teicher Identifiability of finite mixtures, Ann. Math. Statist., Tome 34 (1963), pp. 1265-1269 | Article | MR 155376 | Zbl 0137.12704

[17] D.M. Titterington; A.F. Smith; U.E. Makov Statistical Analysis of Finite Mixture Distributions, John Wiley \& Sons (1985) | MR 838090 | Zbl 0646.62013

[18] H. Weyl The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton N.J. (1939) | MR 1488158 | Zbl 0020.20601

[19] S.J. Yakowitz; J. D. Spragins On the identifiability of finite mixtures, Ann. Math. Statist., Tome 39 (1968), pp. 209-214 | Article | MR 224204 | Zbl 0155.25703