A Semidefinite Relaxation Method for Partially Symmetric Tensor Decomposition

References

• [1] Ballico E, Bernardi A, Christandl M, Gesmundo F (2019) On the partially symmetric rank of tensor products of W-states and other symmetric tensors. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30(1):93–124.Crossref, Google Scholar
• [2] Bernardi A, Brachat J, Comon P, Mourrain B (2013) General tensor decomposition, moment matrices and applications. J. Symbolic Comput. 52:51–71.Crossref, Google Scholar
• [3] Brachat J, Comon P, Mourrain B, Tsigaridas E (2010) Symmetric tensor decomposition. Linear Algebra Appl. 433(11-12):1851–1872.Crossref, Google Scholar
• [4] Breiding P, Vannieuwenhoven N (2018) A Riemannian trust region method for the canonical tensor rank approximation problem. SIAM J. Optim. 28(3):2435–2465.Crossref, Google Scholar
• [5] Comon P, Golub G, Lim L-H, Mourrain B (2008) Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl. 30(3):1254–1279.Crossref, Google Scholar
• [6] Comon P, Lim L-H, Qi Y, Ye K (2020) Topology of tensor ranks. Adv. Math. 367:1–46.Crossref, Google Scholar
• [7] Curto R, Fialkow L (2005) Truncated K-moment problems in several variables. J. Operator Theory 54(1):189–226.Google Scholar
• [8] De Lathauwer L (2006) A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization. SIAM J. Matrix Anal. Appl. 28(3):642–666.Crossref, Google Scholar
• [9] De Lathauwer L, De Moor B, Vandewalle J (2004) Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition. SIAM J. Matrix Anal. Appl. 26(2):295–327.Crossref, Google Scholar
• [10] Dressler M, Nie J, Yang Z (2021) Separability of Hermitian tensors and PSD decompositions. Linear Multilinear Algebra, ePub ahead of print August 27, https://doi.org/10.1080/03081087.2021.1965078.Crossref, Google Scholar
• [11] Fan J, Nie J, Zhou A (2019) Completely positive binary tensors. Math. Oper. Res. 44(3):1087–1100.Link, Google Scholar
• [12] Fialkow L, Nie J (2012) The truncated moment problem via homogenization and flat extensions. J. Funct. Anal. 263(6):1682–1700.Crossref, Google Scholar
• [13] Gesmundo F, Oneto A, Ventura E (2019) Partially symmetric variants of Comon’s problem via simultaneous rank. SIAM J. Matrix Anal. Appl. 40(4):1453–1477.Crossref, Google Scholar
• [14] Henrion D, Lasserre J, Loefberg J (2009) GloptiPoly 3: Moments, optimization and semidefinite programming. Optim. Methods Software 24(4-5):761–779.Crossref, Google Scholar
• [15] Hillar C, Lim L-H (2013) Most tensor problems are NP-hard. J. ACM 60(6):1–39.Crossref, Google Scholar
• [16] Kolda TG, Bader BW (2009) Tensor decompositions and applications. SIAM Rev. 51(3):455–500.Crossref, Google Scholar
• [17] Landsberg JM (2012) Tensors: Geometry and Applications, Graduate Studies in Mathematics, vol. 128 (American Mathematical Society, Providence, RI).Google Scholar
• [18] Landsberg JM, Teitler Z (2010) On the ranks and border ranks of symmetric tensors. Foundations Comput. Math. 10:339–366.Crossref, Google Scholar
• [19] Lasserre JB (2001) Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3):796–817.Crossref, Google Scholar
• [20] Li Y, Ni G (2020) Separability discrimination and decomposition of m-partite quantum mixed states. Phys. Rev. A 102:012402.Crossref, Google Scholar
• [21] Ni G, Yang B (2021) Symmetric Hermitian decomposability criterion, decomposition and its applications. Frontiers Math. China, ePub ahead of print November 6, https://doi.org/10.1007/s11464-021-0927-4.Crossref, Google Scholar
• [22] Nie J (2013) An exact Jacobian SDP relaxation for polynomial optimization. Math. Programming 137:225–255.Crossref, Google Scholar
• [23] Nie J (2014) The A-truncated K-moment problem. . Foundations Comput. Math. 14(6):1243–1276.Crossref, Google Scholar
• [24] Nie J (2015) The hierarchy of local minimums in polynomial optimization. Math. Programming 151:555–583.Crossref, Google Scholar
• [25] Nie J, Yang Z (2020) Hermitian tensor decompositions. SIAM J. Matrix Anal. Appl. 41(3):1115–1144.Crossref, Google Scholar
• [26] Putinar M (1993) Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3):969–984.Crossref, Google Scholar
• [27] Seigal A (2020) Ranks and symmetric ranks of cubic surfaces. J. Symbolic Comput. 101:304–317.Crossref, Google Scholar
• [28] Sidiropoulos ND, Bro R (2000) On the uniqueness of multilinear decomposition of N-way arrays. J. Chemometrics 14(3):229–239.Crossref, Google Scholar
• [29] Sturm JF (1999) Using SeDuMi 1.02: AMATLAB toolbox for optimization over symmetric cones. Optim. Methods Software 11(1-4):625–653.Crossref, Google Scholar
• [30] Wang X, Che M, Wei Y (2018) Best rank-one approximation of fourth-order partially symmetric tensors by neural network. Numerical Math. Theory Methods Appl. 11(4):673–700.Crossref, Google Scholar
• [31] Zhang X, Huang ZH, Qi L (2016) Comon’s conjecture, rank decomposition, and symmetric rank decomposition of symmetric tensors. SIAM J. Matrix Anal. Appl. 37(4):1719–1728.Crossref, Google Scholar
• [32] Zhao X, Fan J (2019) A semidefinite method for tensor complementarity problems. Optim. Methods Software 34(4):758–769.Crossref, Google Scholar