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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| Published in | Foundations of computational mathematics Vol. 10; no. 3; pp. 339 - 366 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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Springer-Verlag
01.06.2010
Springer Springer Nature B.V |
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| Abstract | 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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| AbstractList | 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. Keywords Symmetric tensor rank * Border rank * Secant varieties Mathematics Subject Classification (2000) 15A21 * 15A69 * 14N15 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. 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. [PUBLICATION ABSTRACT] |
| Audience | Academic |
| Author | Teitler, Zach Landsberg, J. M. |
| Author_xml | – sequence: 1 givenname: J. M. surname: Landsberg fullname: Landsberg, J. M. email: jml@math.tamu.edu organization: Department of Mathematics, Texas A&M University – sequence: 2 givenname: Zach surname: Teitler fullname: Teitler, Zach organization: Department of Mathematics, Texas A&M University |
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| Title | On the Ranks and Border Ranks of Symmetric Tensors |
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