Ranks of tensors and a generalization of secant varieties

Abstract

We introduce subspace rank as a tool for studying ranks of tensors and X-rank more generally. We derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in C^2 \otimes C^b \otimes C^b. We review the literature from a geometric perspective.

22 pages; final published version; Linear Algebra and its Applications 2012

References in corpus (2)

• Computing symmetric rank for symmetric tensors
• Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture

Cited by in corpus (31)

• Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
• An algorithm for generic and low-rank specific identifiability of complex tensors
• Effective criteria for specific identifiability of tensors and forms
• Stratification of the fourth secant variety of Veronese variety via the symmetric rank
• Waring loci and the Strassen conjecture
• Tensor ranks on tangent developable of Segre varieties
• Geometric lower bounds for generalized ranks
• Four lectures on secant varieties
• Partially symmetric variants of Comon's problem via simultaneous rank
• An Algebraic-Geometric Characterization of Tripartite Entanglement
• A link between Quantum Entanglement, Secant varieties and Sphericity
• Grassmann secants, identifiability, and linear systems of tensors
• Border Ranks of Monomials
• On the typical rank of real binary forms
• A Gap in the Subrank of Tensors
• Computations and Equations for Segre-Grassmann hypersurfaces
• Unique decomposition for a polynomial of low rank
• word2ket: Space-efficient Word Embeddings inspired by Quantum Entanglement
• Hackbusch Conjecture on tensor formats
• The average condition number of most tensor rank decomposition problems is infinite
• Maximum Waring ranks of monomials
• Are all Secant Varieties of Segre Products Arithmetically Cohen-Macaulay?
• Border rank non-additivity for higher order tensors
• On tensors of border rank $l$ in $\C^{m\times n\times l}$
• Classifying Entanglement by Algebraic Geometry
• Fast structured matrix computations: tensor rank and Cohn--Umans method
• Curvilinear schemes and maximum rank of forms
• Nondefective secant varieties of varieties of completely decomposable forms
• Hilbert Functions of Chopped Ideals
• On Comon's and Strassen's conjectures
• Sets computing the symmetric tensor rank