Algebraic and Computational Aspects of Real Tensor Ranks
This book provides comprehensive summaries of theoretical (algebraic) and computational aspects of tensor ranks, maximal ranks, and typical ranks, over the real number field. Although tensor ranks have been often argued in the complex number field, it should be emphasized that this book treats real tensor ranks, which have direct applications in statistics. The book provides several interesting ideas, including determinant polynomials, determinantal ideals, absolutely nonsingular tensors, absolutely full column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. In addition to reviews of methods to determine real tensor ranks in details, global theories such as the Jacobian method are also reviewed in details. The book includes as well an accessible and comprehensive introduction of mathematical backgrounds, with basics of positive polynomials and calculations by using the Groebner basis. Furthermore, this book provides insights into numerical methods of finding tensor ranks through simultaneous singular value decompositions.
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3-tensor absolutely nonsingular tensor affine algebraic variety Algebraic and Computational AT'B bilinear map characteristic polynomial consider Corollary define Definition denote dense open subset det(A diagonal matrices diagonalizable element Euclidean open set exists family of order following fact format f full column rank GL(m GL(n grank(f Hurwitz–Radon family infinite field integer irreducible affine algebraic JSS Research Series Kronecker product Kronecker–Weierstrass Lemma Let A1 Let f linearly independent max.rank max.rankHz max.rankk(m maximal rank minimal typical rank nonzero number field open set open subset positive integer Proof Let rank of tensors rank of Tp(f rank-1 tensors rank(T rankk(A rankš rankp rational function rational maps real number Real Tensor Ranks resp Sakata semi-algebraic sets Series in Statistics Strassen Sumi Suppose Tc(f tensor rank tensor with format Theorem Tin(f topology vector space x e domp Zariski zero