On the Ranks and Border Ranks of Symmetric Tensors

Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by consideri...

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Bibliographic Details
Published inFoundations of computational mathematics Vol. 10; no. 3; pp. 339 - 366
Main Authors Landsberg, J. M., Teitler, Zach
Format Journal Article
LanguageEnglish
Published New York Springer-Verlag 01.06.2010
Springer
Springer Nature B.V
Subjects
Applications of Mathematics
Computer Science
Economics
Geometry
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Polynomials
Border rank
15A69
14N15
Secant varieties
Symmetric tensor rank
15A21
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Summary:Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-009-9055-3