×

A normal form algorithm for tensor rank decomposition. (English) Zbl 07908552

MSC:

65-XX Numerical analysis

Software:

Macaulay2

References:

[1] Beltrán, C., Breiding, P., and Vannieuwenhoven, N.. 2019. Pencil-based algorithms for tensor rank decomposition are not stable. SIAM J. Matrix Anal. Appl.40, 2 (2019), 739-773. · Zbl 1451.14170
[2] Bender, M. R. and Telen, S.. 2020. Toric eigenvalue methods for solving sparse polynomial systems. Retrieved from https://arXiv:2006.10654.
[3] Bernardi, A., Brachat, J., Comon, P., and Mourrain, B.. 2011. Multihomogeneous polynomial decomposition using moment matrices. In Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation. 35-42. · Zbl 1323.15017
[4] Bernardi, A., Brachat, J., Comon, P., and Mourrain, B.. 2013. General tensor decomposition, moment matrices and applications. J. Symbolic Comput.52 (2013), 51-71. · Zbl 1275.15017
[5] Bernardi, A. and Taufer, D.. 2020. Waring, tangential and cactus decompositions. J. Math. Pures Appl.143 (2020), 1-30. · Zbl 1448.14051
[6] Bocci, C., Chiantini, L., and Ottaviani, G.. 2014. Refined methods for the identifiability of tensors. Ann. Mat. Pura Appl.193, 6 (2014), 1691-1702. · Zbl 1314.14102
[7] Brachat, J., Comon, P., Mourrain, B., and Tsigaridas, E.. 2010. Symmetric tensor decomposition. Linear Alg. Appl.433, 11-12 (2010), 1851-1872. · Zbl 1206.65141
[8] Bunch, J. R. and Kaufman, L.. 1977. Some stable methods for calculating inertia and solving symmetric linear systems. Math. Comp.31 (1977), 163-179. · Zbl 0355.65023
[9] Bürgisser, P., Clausen, M., and Shokrollahi, A.. 1997. Algebraic Complexity Theory. , Vol. 315. Springer-Verlag, Berlin. · Zbl 1087.68568
[10] Carlini, E. and Kleppe, J.. 2011. Ranks derived from multilinear maps. J. Pure Appl. Alg.215, 8 (2011), 1999-2004. · Zbl 1215.15029
[11] Chiantini, L. and Ciliberto, C.. 2006. On the concept of \(k \) -secant order of a variety. J. London Math. Soc.73, 2 (2006), 436-454. · Zbl 1101.14067
[12] Chiantini, L. and Ottaviani, G.. 2012. On generic identifiability of 3-tensors of small rank. SIAM J. Matrix Anal. Appl.33, 3 (2012), 1018-1037. · Zbl 1263.14053
[13] Chiantini, L., Ottaviani, G., and Vannieuwenhoven, N.. 2014. An algorithm for generic and low-rank specific identifiability of complex tensors. SIAM J. Matrix Anal. Appl.35, 4 (2014), 1265-1287. · Zbl 1322.14022
[14] Chiantini, L., Ottaviani, G., and Vannieuwenhoven, N.. 2017. Effective criteria for specific identifiability of tensors and forms. SIAM J. Matrix Anal. Appl.38, 2 (2017), 656-681. · Zbl 1371.65038
[15] Cox, D. A., Little, J., and O’Shea, D.. 2006. Using Algebraic Geometry. , Vol. 185. Springer Science & Business Media.
[16] Cox, D. A., Little, J., and O’Shea, D.. 2013. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer Science & Business Media.
[17] Cox, D. A., Little, J., and Schenck, H. K.. 2011. Toric Varieties. Vol. 124. American Mathematical Society. · Zbl 1223.14001
[18] Lathauwer, L. De. 2006. A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization. SIAM J. Matrix Anal. Appl.28, 3 (2006), 642-666. · Zbl 1126.15007
[19] Lathauwer, L. De, Moor, B. De, and Vandewalle, J.. 2000. A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl.21, 4 (2000), 1253-1278. · Zbl 0962.15005
[20] Domanov, I. and Lathauwer, L. De. 2013. 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 (2013), 876-903. · Zbl 1282.15020
[21] Domanov, I. and Lathauwer, L. De. 2014. Canonical polyadic decomposition of third-order tensors: Reduction to generalized eigenvalue decomposition. SIAM J. Matrix Anal. Appl.35, 2 (2014), 636-660. · Zbl 1306.15022
[22] Domanov, I. and Lathauwer, L. De. 2017. Canonical polyadic decomposition of third-order tensors: Relaxed uniqueness conditions and algebraic algorithm. Linear Alg. Appl.513 (2017), 342-375. · Zbl 1349.15065
[23] Grayson, D. R. and Stillman, M. E.. 2019. Macaulay2, a software system for research in algebraic geometry. software version: v1.18. Retrieved from http://www.math.uiuc.edu/Macaulay2/.
[24] Greub, W.. 1978. Multilinear Algebra (2nd ed.). Springer-Verlag. · Zbl 0387.15001
[25] Hendrikx, S., Boussé, M., Vervliet, N., Vandecappelle, M., Kenis, R., and Lathauwer, L. De. 2022. Tensorlab^+. Retrieved from https://www.tensorlabplus.net.
[26] Hillar, C. J. and Lim, L.-H.. 2013. Most tensor problems are NP-Hard. J. ACM60, 6 (2013), 45:1-45:39. · Zbl 1281.68126
[27] Hitchcock, F. L.. 1927. The expression of a tensor or a polyadic as a sum of products. J. Math. Phys.6, 1 (1927), 164-189. · JFM 53.0095.01
[28] Iarrobino, Anthony and Kanev, Vassil. 1999. Power Sums, Gorenstein Algebras, and Determinantal Loci. Springer Science & Business Media. · Zbl 0942.14026
[29] Kolda, T. G. and Bader, B. W.. 2009. Tensor decompositions and applications. SIAM Rev.51, 3 (2009), 455-500. · Zbl 1173.65029
[30] Kruskal, J. B.. 1977. Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Alg. Appl.18 (1977), 95-138. · Zbl 0364.15021
[31] Kuo, Y.-Ch. and Lee, T.-L.. 2018. Computing the unique CANDECOMP/PARAFAC decomposition of unbalanced tensors by homotopy method. Linear Alg. Appl.556 (2018), 238-264. · Zbl 1395.15018
[32] Landsberg, J. M.. 2012. Tensors: Geometry and Applications. , Vol. 128. AMS, Providence, RI. · Zbl 1238.15013
[33] Leurgans, S. E., Ross, R. T., and Abel, R. B.. 1993. A decomposition for three-way arrays. SIAM J. Matrix Anal. Appl.14, 4 (1993), 1064-1083. · Zbl 0788.65145
[34] Lorber, A.. 1985. Features of quantifying chemical composition from two-dimensional data array by the rank annihilation factor analysis method. Anal. Chem.57 (1985), 2395-2397.
[35] Macaulay, F. S.. 1916. The Algebraic Theory of Modular Systems. Vol. 19. Cambridge University Press. · JFM 46.0167.01
[36] Maclagan, D. and Smith, G. G.. 2004. Multigraded castelnuovo-mumford regularity. J. für die Reine und Angew. Math.2004, 571 (2004), 179-212. · Zbl 1062.13004
[37] Miller, E. and Sturmfels, B.. 2005. Combinatorial Commutative Algebra. Vol. 227. Springer Science & Business Media. · Zbl 1090.13001
[38] Mourrain, B.. 2018. Polynomial-exponential decomposition from moments. Found. Comput. Math.18, 6 (2018), 1435-1492. · Zbl 1427.14119
[39] Nie, J.. 2017. Generating polynomials and symmetric tensor decompositions. Found. Comput. Math.17, 2 (2017), 423-465. · Zbl 1381.15017
[40] Oseledets, I. V., Savostianov, D. V., and Tyrtyshnikov, E. E.. 2008. Tucker dimensionality reduction of three-dimensional arrays in linear time. SIAM J. Matrix Anal. Appl.30, 3 (2008), 939-956. · Zbl 1180.15025
[41] Russo, Francesco. 2016. On the Geometry of Some Special Projective Varieties. Springer. · Zbl 1337.14001
[42] Sanchez, E. and Kowalski, B. R.. 1990. Tensorial resolution: A direct trilinear decomposition. J. Chemom.4, 1 (1990), 29-45.
[43] Sidiropoulos, N. D. and Bro, R.. 2000. On the uniqueness of multilinear decomposition of N-way arrays. J. Chemom.14, 3 (2000), 229-239.
[44] Sidiropoulos, N. D., Lathauwer, L. De, Fu, X., Huang, K., Papalexakis, E. E., and Faloutsos, Ch.. 2017. Tensor decomposition for signal processing and machine learning. IEEE Trans. Signal. Process.65, 13 (2017), 3551-3582. · Zbl 1415.94232
[45] Telen, S.. 2020a. Numerical root finding via cox rings. J. Pure Appl. Alg.224, 9 (2020), 106367. · Zbl 1442.14187
[46] Telen, S.. 2020b. Solving Systems of Polynomial Equations. . KU Leuven.
[47] Telen, S., Mourrain, B., and Barel, M. Van. 2018. Solving polynomial systems via truncated normal forms. SIAM J. Matrix Anal. Appl.39, 3 (2018), 1421-1447. · Zbl 1401.65054
[48] Tucker, L. R.. 1966. Some mathematical notes on three-mode factor analysis. Psychometrika31, 3 (1966), 279-311.
[49] Vannieuwenhoven, N., Vandebril, R., and Meerbergen, K.. 2012. A new truncation strategy for the higher-order singular value decomposition. SIAM J. Sci. Comput.34, 2 (2012), A1027-A1052. · Zbl 1247.65055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.