From non-defectivity to identifiability. (English) Zbl 07683503

Casarotti, Alex; Mella, Massimiliano

J. Eur. Math. Soc. (JEMS) 25, No. 3, 913-931 (2023).

Summary: A point \(p\) in a projective space is \(h\)-identifiable via a variety \(X\) if there is a unique way to write \(p\) as a linear combination of \(h\) points of \(X\). Identifiability is important both in algebraic geometry and in applications. In this paper we propose an entirely new approach to study identifiability, connecting it to the notion of secant defect for any smooth projective variety. In this way we are able to improve the known bounds on identifiability and produce new identifiability statements. In particular, we give optimal bounds for some Segre and Segre-Veronese varieties and provide the first identifiability statements for Grassmann varieties.

MSC:

14N05	Projective techniques in algebraic geometry
15A69	Multilinear algebra, tensor calculus
15A72	Vector and tensor algebra, theory of invariants
11P05	Waring’s problem and variants
14N07	Secant varieties, tensor rank, varieties of sums of powers

Keywords:

tensor decomposition; Waring decomposition; identifiability; defective varieties

Software:

SeDiMO

References:

[1]	Abo, H., Brambilla, M. C.: Secant varieties of Segre-Veronese varieties P m P n embedded by O.1; 2/. Experiment. Math. 18, 369-384 (2009) Zbl 1198.14051 MR 2555705 · Zbl 1198.14051
[2]	Abo, H., Ottaviani, G., Peterson, C.: Induction for secant varieties of Segre varieties. Trans. Amer. Math. Soc. 361, 767-792 (2009) Zbl 1170.14036 MR 2452824 · Zbl 1170.14036
[3]	Alexander, J., Hirschowitz, A.: La méthode d’Horace éclatée: application à l’interpolation en degré quatre. Invent. Math. 107, 585-602 (1992) Zbl 0784.14002 MR 1150603 · Zbl 0784.14002
[4]	Améndola, C., Faugère, J.-C., Sturmfels, B.: Moment varieties of Gaussian mixtures. J. Algeb-raic Statist. 7, 14-28 (2016) Zbl 1361.13017 MR 3529332 · Zbl 1361.13017
[5]	Améndola, C., Ranestad, K., Sturmfels, B.: Algebraic identifiability of Gaussian mixtures. Int. Math. Res. Notices 2018, 6556-6580 Zbl 1423.14284 MR 3873537 · Zbl 1423.14284
[6]	Araujo, C., Massarenti, A., Rischter, R.: On non-secant defectivity of Segre-Veronese variet-ies. Trans. Amer. Math. Soc. 371, 2255-2278 (2019) Zbl 1404.14062 MR 3896080 · Zbl 1404.14062
[7]	Ballico, E., Bernardi, A., Catalisano, M. V.: Higher secant varieties of P n P 1 embedded in bi-degree .a; b/. Comm. Algebra 40, 3822-3840 (2012) Zbl 1262.14066 MR 2982901 · Zbl 1262.14066
[8]	Ballico, E., Bernardi, A., Chiantini, L.: On the dimension of contact loci and the identifiability of tensors. Ark. Mat. 56, 265-283 (2018) Zbl 1464.14054 MR 3893774 · Zbl 1464.14054
[9]	Baur, K., Draisma, J., de Graaf, W. A.: Secant dimensions of minimal orbits: computations and conjectures. Experiment. Math. 16, 239-250 (2007) Zbl 1162.14038 MR 2339279 · Zbl 1162.14038
[10]	Bernardi, A., Carlini, E., Catalisano, M. V.: Higher secant varieties of P n P m embedded in bi-degree .1; d /. J. Pure Appl. Algebra 215, 2853-2858 (2011) Zbl 1231.14044 MR 2811568 · Zbl 1231.14044
[11]	Bernardi, A., Vanzo, D.: A new class of non-identifiable skew-symmetric tensors. Ann. Mat. Pura Appl. (4) 197, 1499-1510 (2018) Zbl 1403.15017 MR 3848461 · Zbl 1403.15017
[12]	Bocci, C., Chiantini, L.: On the identifiability of binary Segre products. J. Algebraic Geom. 22, 1-11 (2013) Zbl 1273.14107 MR 2993044 · Zbl 1273.14107
[13]	Bocci, C., Chiantini, L., Ottaviani, G.: Refined methods for the identifiability of tensors. Ann. Mat. Pura Appl. (4) 193, 1691-1702 (2014) Zbl 1314.14102 MR 3275258 · Zbl 1314.14102
[14]	Catalisano, M. V., Geramita, A. V., Gimigliano, A.: Secant varieties of P 1 P 1 (n-times) are not defective for n 5. J. Algebraic Geom. 20, 295-327 (2011) Zbl 1217.14039 MR 2762993 · Zbl 1217.14039
[15]	Chiantini, L.: private communication
[16]	Chiantini, L., Ciliberto, C.: Weakly defective varieties. Trans. Amer. Math. Soc. 354, 151-178 (2002) Zbl 1045.14022 MR 1859030 · Zbl 1045.14022
[17]	Chiantini, L., Ciliberto, C.: On the dimension of secant varieties. J. Eur. Math. Soc. 12, 1267-1291 (2010) Zbl 1201.14038 MR 2677616 · Zbl 1201.14038
[18]	Chiantini, L., Mella, M., Ottaviani, G.: One example of general unidentifiable tensors. J. Algebraic Statist. 5, 64-71 (2014) Zbl 1346.14125 MR 3279954 · Zbl 1346.14125
[19]	Chiantini, L., Ottaviani, G.: On generic identifiability of 3-tensors of small rank. SIAM J. Matrix Anal. Appl. 33, 1018-1037 (2012) Zbl 1263.14053 MR 3023462 · Zbl 1263.14053
[20]	Chiantini, L., Ottaviani, G., Vannieuwenhoven, N.: An algorithm for generic and low-rank specific identifiability of complex tensors. SIAM J. Matrix Anal. Appl. 35, 1265-1287 (2014) Zbl 1322.14022 MR 3270978 · Zbl 1322.14022
[21]	Chiantini, L., Ottaviani, G., Vannieuwenhoven, N.: On generic identifiability of symmetric tensors of subgeneric rank. Trans. Amer. Math. Soc. 369, 4021-4042 (2017) Zbl 1360.14021 MR 3624400 · Zbl 1360.14021
[22]	Domanov, I., De Lathauwer, L.: On the uniqueness of the canonical polyadic decomposition of third-order tensors-Part I: Basic results and uniqueness of one factor matrix. SIAM J. Matrix Anal. Appl. 34, 855-875 (2013) Zbl 1282.15019 MR 3072760 · Zbl 1282.15019
[23]	Domanov, I., De Lathauwer, L.: On the uniqueness of the canonical polyadic decomposition of third-order tensors-Part II: Uniqueness of the overall decomposition. SIAM J. Matrix Anal. Appl. 34, 876-903 (2013) Zbl 1282.15020 MR 3072761 · Zbl 1282.15020
[24]	Domanov, I., De Lathauwer, L.: Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL. SIAM J. Matrix Anal. Appl. 36, 1567-1589 (2015) Zbl 1330.15028 MR 3421620 · Zbl 1330.15028
[25]	Galuppi, F., Mella, M.: Identifiability of homogeneous polynomials and Cremona transform-ations. J. Reine Angew. Math. 757, 279-308 (2019) Zbl 1437.14023 MR 4036576 · Zbl 1437.14023
[26]	Griffiths, P., Harris, J.: Algebraic geometry and local differential geometry. Ann. Sci. École Norm. Sup. (4) 12, 355-452 (1979) Zbl 0426.14019 MR 559347 · Zbl 0426.14019
[27]	Hilbert, D.: Lettre adressée à M. Hermite, Gesam. Abh., Vol. II, 148-153
[28]	Karfoul, A., Albera, L., De Lathauwer, L.: Iterative methods for the canonical decomposi-tion of multi-way arrays: Application to blind underdetermined mixture identification. Signal Processing 91, 1789-1802 (2011) Zbl 1217.94051 · Zbl 1217.94051
[29]	Kruskal, J. B.: Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra Appl. 18, 95-138 (1977) Zbl 0364.15021 MR 444690 · Zbl 0364.15021
[30]	Laface, A., Postinghel, E.: Secant varieties of Segre-Veronese embeddings of .P 1 / r . Math. Ann. 356, 1455-1470 (2013) Zbl 1275.14041 MR 3072808 · Zbl 1275.14041
[31]	Lickteig, T.: Typical tensorial rank. Linear Algebra Appl. 69, 95-120 (1985) Zbl 0575.15013 MR 798367 · Zbl 0575.15013
[32]	Massarenti, A., Rischter, R.: Non-secant defectivity via osculating projections. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 19, 1-34 (2019) Zbl 1419.14080 MR 3923838 · Zbl 1419.14080
[33]	Mella, M.: Singularities of linear systems and the Waring problem. Trans. Amer. Math. Soc. 358, 5523-5538 (2006) Zbl 1112.14062 MR 2238925 · Zbl 1112.14062
[34]	Sidiropoulos, N. D., Bro, R.: On the uniqueness of multilinear decomposition of N -way arrays. J. Chemometrics 14, 229-239 (2000)
[35]	Sylvester, J. J.: Collected Works. Cambridge Univ. Press (1904)
[36]	Vannieuwenhoven, N., Vandebril, R., Meerbergen, K.: A randomized algorithm for testing nonsingularity of structured matrices with an application to asserting nondefectivity of Segre varieties. IMA J. Numer. Anal. 35, 289-324 (2015) Zbl 1320.65067 MR 3335206 · Zbl 1320.65067