The Hitchhiker guide to: secant varieties and tensor decomposition

DOI10.3390/math6120314zbMath1425.14043arXiv1812.10267OpenAlexW2904001778WikidataQ128703687 ScholiaQ128703687MaRDI QIDQ2337279

Alessandro Oneto, Enrico Carlini, Alessandro Gimigliano, Alessandra Bernardi, Maria Virginia Catalisano

Publication date: 19 November 2019

Published in: Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1812.10267

zbMATH Keywords

algorithmGrassmanniansSegre varietiestensor ranksecant varietiesWaring rankVeronese varietiesSegre-Veronese varietiesadditive decompositions

Mathematics Subject Classification ID

Vector and tensor algebra, theory of invariants (15A72) Research exposition (monographs, survey articles) pertaining to algebraic geometry (14-02) Projective techniques in algebraic geometry (14N05)

Related Items (max. 100)

Strict inclusions of high rank loci ⋮ Geometric conditions for strict submultiplicativity of rank and border rank ⋮ A bound for the Waring rank of the determinant via syzygies ⋮ Border rank of powers of ternary quadratic forms ⋮ The Rank of Trifocal Grassmann Tensors ⋮ The monic rank ⋮ More on Tensors with Different Rank and Symmetric Rank ⋮ Waring, tangential and cactus decompositions ⋮ Waring rank of symmetric tensors, and singularities of some projective hypersurfaces ⋮ On minimal decompositions of low rank symmetric tensors ⋮ Identifiability of rank-3 tensors ⋮ Tangential varieties of Segre-Veronese surfaces are never defective ⋮ Special apolar subset: the case of star configurations ⋮ All secant varieties of the Chow variety are nondefective for cubics and quaternary forms ⋮ Waring rank of binary forms, harmonic cross-ratio and golden ratio ⋮ Secant non-defectivity via collisions of fat points

Uses Software

• SeDiMO

Cites Work

• The asymptotic leading term for maximum rank of ternary forms of a given degree
• Real and complex Waring rank of reducible cubic forms
• Some examples of forms of high rank
• Equations for secant varieties of Veronese and other varieties
• Waring decompositions of monomials
• Eigenvectors of tensors and algorithms for Waring decomposition
• A comparison of different notions of ranks of symmetric tensors
• Decomposition of homogeneous polynomials with low rank
• An asymptotic bound for secant varieties of Segre varieties
• Refined methods for the identifiability of tensors
• Sufficient conditions for Strassen's additivity conjecture
• Symmetric tensor decomposition
• On non-defectivity of certain Segre-Veronese varieties
• Computing symmetric rank for symmetric tensors
• On the variety parameterizing completely decomposable polynomials
• On the rank of a binary form
• On the generic and typical ranks of 3-tensors
• Higher secant varieties of \(\mathbb P^n \times \mathbb P^n\) embedded in bi-degree \((1,d)\)
• On the direct sum conjecture
• Special effect varieties in higher dimension
• Varieties with minimal secant degree and linear systems of maximal dimension on surfaces
• A probabilistic algorithm for the secant defect of Grassmann varieties
• Regularity of linear systems of plane curves
• Tensor-based techniques for the blind separation of DS-CDMA signals
• On the ranks and border ranks of symmetric tensors
• Secant varieties to osculating varieties of Veronese embeddings of \(\mathbb P^n\)
• Identifiability of parameters in latent structure models with many observed variables
• Rank and optimal computation of generic tensors
• La methode d'Horace pour l'interpolation à plusieurs variables
• Typical tensorial rank
• Generic hypersurface singularities
• Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics
• Irreducible lengths of trivectors of rank seven and eight
• Syzygies des variétés déterminantales
• Apolarity and canonical forms for homogeneous polynomials
• The Veronese variety and catalecticant matrices
• Fat points, inverse systems, and piecewise polynomial functions
• The red book of varieties and schemes. Includes the Michigan lectures (1974) on ``Curves and their Jacobians.
• Chordal varieties of Veronese varieties and catalecticant matrices
• Fat points on rational normal curves
• Independent component analysis, a new concept?
• On the secant varieties to the osculating variety of a Veronese surface
• On Waring's problem for several algebraic forms
• An asymptotic vanishing theorem for generic unions of multiple points
• Gram determinants of real binary tensors
• On the number of Waring decompositions for a generic polynomial vector
• Spherical blow-ups of Grassmannians and Mori dream spaces
• On generic and maximal \(k\)-ranks of binary forms
• A note on the cactus rank for Segre-Veronese varieties
• On the locus of points of high rank
• Varieties of apolar subschemes of toric surfaces
• A new class of non-identifiable skew-symmetric tensors
• On the ideals of secant varieties of Segre varieties
• The geometric interpretation of Fröberg-Iarrobino conjectures on infinitesimal neighbourhoods of points in projective space
• Power sums, Gorenstein algebras, and determinantal loci. With an appendix `The Gotzmann theorems and the Hilbert scheme' by Anthony Iarrobino and Steven L. Kleiman
• Stratified exponential families: Graphical models and model selection
• Ranks of tensors, secant varieties of Segre varieties and fat points
• On the dimensions of secant varieties of Segre-Veronese varieties
• The solution to the Waring problem for monomials and the sum of coprime monomials
• New examples of defective secant varieties of Segre-Veronese varieties
• Multivariate polynomials, duality, and structured matrices
• On real typical ranks
• Counterexamples to Strassen's direct sum conjecture
• Identifiability of homogeneous polynomials and Cremona transformations
• On minimal decompositions of low rank symmetric tensors
• Varieties of completely decomposable forms and their secants
• Progress on the symmetric Strassen conjecture
• Typical real ranks of binary forms
• Waring loci and the Strassen conjecture
• Real rank geometry of ternary forms
• Secant varieties of Segre-Veronese embeddings of \((\mathbb{P }^1)^r\)
• On the real rank of monomials
• Abelian tensors
• Waring-like decompositions of polynomials. I.
• Secant varieties of the varieties of reducible hypersurfaces in \(\mathbb{P}^n\)
• On the ideals of secant varieties to certain rational varieties
• On the rank of a symmetric form
• \(3 \times 3\) minors of catalecticants
• An upper bound for the Waring rank of a form
• Algebraic geometry of Bayesian networks
• A tropical approach to secant dimensions
• Phylogenetic ideals and varieties for the general Markov model
• On the Alexander-Hirschowitz theorem
• On the secant varieties to the tangent developable of a Veronese variety
• Some defective secant varieties to osculating varieties of Veronese surfaces
• On the non-defectivity of non weak-defectivity of Segre-Veronese embeddings of products of projective spaces
• On maximum, typical and generic ranks
• Higher secant varieties of the Segre varieties \(\mathbb P^1\times\dots\times\mathbb P^1\)
• Segre-Veronese embeddings of \(\mathbb P^1\times\mathbb P^1\times\mathbb P^1\) and their secant varieties
• A brief proof of a maximal rank theorem for generic double points in projective space
• Weakly defective varieties
• On the secant varieties to the tangential varieties of a Veronesean
• Remarks on the Symmetric Rank of Symmetric Tensors
• A Proof that the Maximal Rank for Plane Quartics is Seven
• Comon's Conjecture, Rank Decomposition, and Symmetric Rank Decomposition of Symmetric Tensors
• Stratification of the fourth secant variety of Veronese varieties via the symmetric rank
• Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture
• Tensor ranks on tangent developable of Segre varieties
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Algebraic geometry tools for the study of entanglement: an application to spin squeezed states
• On the typical rank of real binary forms
• Tensor decompositions for learning latent variable models
• On generic identifiability of symmetric tensors of subgeneric rank
• Generic forms of low Chow rank
• Secant varieties of ℙ¹×⋯×ℙ¹ (𝕟-times) are NOT defective for 𝕟≥5
• Monomials as sums of powers: The real binary case
• Non-defectivity of Grassmannians of planes
• SECANTS TO THE VARIETY OF COMPLETELY REDUCIBLE FORMS AND THE HILBERT FUNCTION OF THE UNION OF STAR-CONFIGURATIONS
• Secant Varieties of Segre–Veronese Varieties ℙ^m× ℙⁿEmbedded byO(1, 2)
• Geometry and the complexity of matrix multiplication
• Secant Dimensions of Minimal Orbits: Computations and Conjectures
• About the Defectivity of Certain Segre–Veronese Varieties
• Combinatorics and geometry of power ideals
• On the ideals and singularities of secant varieties of Segre varieties
• Induction for secant varieties of Segre varieties
• An inequality for Hilbert series of graded algebras.
• On the Validity of the Direct Sum Conjecture
• Varieties with an extremal number of degenerate higher secant varieties.
• Degree of the dual of a grassman variety
• Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques.
• The Geometry of Schemes
• Linear systems of plane curves with base points of equal multiplicity
• Varieties of sums of power
• A Counterexample to Comon's Conjecture
• Curvilinear schemes and maximum rank of forms
• On complex and real identifiability of tensors
• Most secant varieties of tangential varieties to Veronese varieties are nondefective
• Symmetric tensors: rank, Strassen's conjecture and e-computability
• Real identifiability vs. complex identifiability
• Secant varieties of Grassmann varieties
• Curves and Symmetric Spaces, I
• The Possible Dimensions of the Higher Secant Varieties
• On Generic Identifiability of 3-Tensors of Small Rank
• Higher Secant Varieties of ℙⁿ × ℙ¹Embedded in Bi-Degree (a,b)
• On the identifiability of binary Segre products
• On the Waring problem for polynomial rings
• On the Hilbert series of ideals generated by generic forms
• Skew-symmetric tensor decomposition
• Quantum computers that can be simulated classically in polynomial time
• Partially Symmetric Variants of Comon's Problem Via Simultaneous Rank
• Small subalgebras of polynomial rings and Stillman’s Conjecture
• Polynomials and tensors of bounded strength
• An Algorithm For Generic and Low-Rank Specific Identifiability of Complex Tensors
• ON DIFFERENCES BETWEEN THE BORDER RANK AND THE SMOOTHABLE RANK OF A POLYNOMIAL
• Every ternary quintic is a sum of ten fifth powers
• Four Lectures on Secant Varieties
• Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties
• Tensor Rank and the Ill-Posedness of the Best Low-Rank Approximation Problem
• Symmetric Tensors and Symmetric Tensor Rank
• On rational normal curves in projective space
• Geometry and Complexity Theory
• On the Length of Binary Forms
• Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes
• Generalizations of Strassen's Equations for Secant Varieties of Segre Varieties
• Variations on Nagata's Conjecture
• On the 14-th Problem of Hilbert

This page was built for publication: The Hitchhiker guide to: secant varieties and tensor decomposition