본문 바로가기
KERD
논문KCI 등재

Beyond the cactus rank of tensors

Edoardo Ballico
대한수학회 (The Korean Mathematical Society)|2018.09|대한수학회보 (Bulletin of the KMS)|제55권 제5호|pp. 1587-1598

1

조회

0

다운로드

0

인용

0

참고문헌

키워드

tensor ranktensor decompositioncactus rankzero-dimensional schemeSegre embedding
주제분류수학

초록

We study additive decompositions (and generalized additive decompositions with a zero-dimensional scheme instead of a finite sum of rank $1$ tensors), which are not of minimal degree (for sums of rank $1$ tensors with more terms than the rank of the tensor, for a zero-dimensional scheme a degree higher than the cactus rank of the tensor). We prove their existence for all degrees higher than the rank of the tensor and, with strong assumptions, higher than the cactus rank of the tensor. Examples show that additional assumptions are needed to get the minimally spanning scheme of degree cactus $+1$.

Abstract

We study additive decompositions (and generalized additive decompositions with a zero-dimensional scheme instead of a finite sum of rank $1$ tensors), which are not of minimal degree (for sums of rank $1$ tensors with more terms than the rank of the tensor, for a zero-dimensional scheme a degree higher than the cactus rank of the tensor). We prove their existence for all degrees higher than the rank of the tensor and, with strong assumptions, higher than the cactus rank of the tensor. Examples show that additional assumptions are needed to get the minimally spanning scheme of degree cactus $+1$.

참고문헌 (31)

  1. E. Ballico, "Tensor ranks on tangent developable of Segre varieties", Linear Multilinear Algebra, 61(7), pp.881~894 (2013)[DOI]
  2. E. Ballico, "Stratification of the fourth secant variety of Veronese varieties via the symmetric rank", Adv. Pure Appl. Math., 4(2), pp.215~250 (2013)
  3. E. Ballico, "Curvilinear schemes and maximum rank of forms", Matematiche (Catania), 72(1), pp.137~144 (2017)
  4. E. Ballico, "On the ranks of the third secant variety of Segre-Veronese embeddings", ~ (2018)
  5. E. Ballico, "On the dimension of contact loci and the identifiability of tensors", ~ (2017)
  6. E. Ballico, "Bounds on the tensor rank", ~
  7. A. Bernardi, "A comparison of different notions of ranks of symmetric tensors", Linear Algebra Appl., 460, pp.205~230 (2014)[DOI]
  8. A. Bernardi, "On the cactus rank of cubics forms", J. Symbolic Comput., 50, pp.291~297 (2013)[DOI]
  9. C. Bocci, "On the identifiability of binary Segre products", J. Algebraic Geom., 22(1), pp.1~11 (2013)
  10. C. Bocci, "Refined methods for the identifiability of tensors", Ann. Mat. Pura Appl., 193(6), pp.1691~1702 (2014)[DOI]
  11. W. Buczynska, "Secant varieties to high degree veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes", J. Algebraic Geom., 23, pp.63~90 (2014)
  12. J. Buczynski, "On the third secant variety", J. Algebraic Combin., 40(2), pp.475~502 (2014)[DOI]
  13. M. V. Catalisano, "Waring-like decompositions of polynomials, 1", Linear Algebra Appl., 533, pp.311~325 (2017)[DOI]
  14. L. Chiantini, "On the concept of k-secant order of a variety", J. London Math. Soc. (2), 73(2), pp.436~454 (2006)[DOI]
  15. L. Chiantini, "On the dimension of secant varieties", J. Eur. Math. Soc. (JEMS), 12(5), pp.1267~1291 (2010)
  16. L. Chiantini, "One example of general unidentifiable tensors", J. Algebr. Stat., 5(1), pp.64~71 (2014)
  17. L. Chiantini, "On generic identifiability of 3-tensors of small rank", SIAM J. Matrix Anal. Appl., 33(3), pp.1018~1037 (2012)[DOI]
  18. L. Chiantini, "An algorithm for generic and low-rank specific identifiability of complex tensors", SIAM J. Matrix Anal. Appl., 35(4), pp.1265~1287 (2014)[DOI]
  19. L. Chiantini, "On generic identifiability of symmetric tensors of subgeneric rank", Trans. Amer. Math. Soc., 369(6), pp.4021~4042 (2017)
  20. L. Chiantini, "Effective criteria for specific identifiability of tensors and forms", SIAM J. Ma- trix Anal. Appl., 38(2), pp.656~681 (2017)[DOI]
  21. H. Derksen, "Kruskal's uniqueness inequality is sharp", Linear Algebra Appl., 438(2), pp.708~712 (2013)[DOI]
  22. I. Domanov, "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(3), pp.855~875 (2013)[DOI]
  23. I. Domanov, "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(3), pp.876~903 (2013)[DOI]
  24. I. Domanov, "Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL", SIAM J. Matrix Anal. Appl., 36(4), pp.1567~1589 (2015)[DOI]
  25. F. Galuppi, "Identifiability of homogeneous polynomials and Cremona Transformations", J. Reine Angew. Math. (2016)[DOI]
  26. A. Iarrobino, "Gorenstein Algebras, and Determinantal Loci", Lecture Notes in Mathematics, 1721 (1999)
  27. J. B. Kruskal, "Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics", Linear Algebra and Appl., 18(2), pp.95~138 (1977)[DOI]
  28. J. M. Landsberg, "Tensors: geometry and applications, Graduate Studies in Mathematics, 128" (2012)
  29. M. Mella, "Singularities of linear systems and the Waring problem", Trans. Amer. Math. Soc., 358(12), pp.5523~5538 (2006)[DOI]
  30. K. Ranestad, "On the rank of a symmetric form", J. Algebra, 346, pp.340~342 (2011)[DOI]
  31. N. D. Sidiropoulos, "On the uniqueness of multilinear decomposition of N-way arrays", J. Chemometrics, 14, pp.229~239 (2000)[DOI]

관련 논문AI추천

Terracini loci of codimension $1$ and a criterion for partially symmetric tensorsEdoardo Ballico대한수학회 · 대한수학회논문집2023 · 제38권 제1호Linearly dependent and concise subsets of a Segre variety depending on $k$ factorsEdoardo Ballico대한수학회 · 대한수학회보2021 · 제58권 제1호Curves with maximal rank, but not aCM, with very high genera in projective spacesEdoardo Ballico대한수학회 · 대한수학회지2019 · 제56권 제5호ON THE EQUATIONS DEFINING SOME CURVES OF MAXIMAL REGULARITY IN P5이완석, Shuailing Yang영남수학회 · East Asian Mathematical Journal2022 · 제38권 제3호Dependent subsets of embedded projective varietiesEdoardo Ballico대한수학회 · 대한수학회보2020 · 제57권 제4호On the equations defining some rational curves of maximal genus in $\mathbb{P}^3$Shuailing Yang, Wanseok Lee영남수학회 · East Asian Mathematical Journal2024 · 제40권 제3호

이 간행물의 다른 논문 (21건)

Height inequality for rational maps and bounds for preperiodic points이종규pp. 1317-1332A QUESTION ON *-REGULAR RINGSJian Cui, Xiaobin Yinpp. 1333-1338Notes on minimal unit Killing vector fields박정형, Kouei Sekigawa, 전선향pp. 1339-1350Polynomially demicompact operators and spectral theory for operator matrices involving demicompactness classesBilel Krichen, Aref Jeribi, Fatma Ben Brahimpp. 1351-1370The classification of self-dual codes over Galois rings of length 4최환혁pp. 1371-1387
원문 보기