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Mathematics
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Decomposability of Tensors
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Decomposability of Tensors

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (28 September 2018) | Viewed by 16754

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Luca Chiantini

E-Mail Website
Guest Editor
Department of Mathematical and Computer Sciences, University of Siena, Pian dei Mantellini, 44, 53100 Siena, Italy
Interests: algebraic geometry; projective geometry; multilinar algebra; commutative algebra; computer algebra; algebraic statistics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Tensor decomposition has recently become a relevant topic, both for theoretical and applied mathematics, due to its interdisciplinary nature, which ranges from multilinear algebra and algebraic geometry to numerical analysis, algebraic statistics, quantum physics, signal processing, artificial intelligence, etc. The starting point behind the study of a decomposition relies on the idea that knowledge of elementary components of a tensor is fundamental to implement procedures able to understand and efficiently handle the information that a tensor encodes. Recent advances started with a systematic application of classical methods (some of them of geometric nature) to determine effective results on tensor decompositions. The methods range from the applications of the geometry of secant varieties in tensor spaces, to the study of symmetries in the decomposition of a specific tensor, to the determination of the sensitivity of a decomposition to small variations (deformations) of the data. Thanks to new applications of theoretic results, criteria for understanding when a given decomposition is minimal or unique, both for generic or specific tensors, have been recently introduced or significantly improved. New types of decompositions, of which elementary blocks can be chosen in a range of different possible models (e.g., Chow decompositions or mixed decompositions) are now systematically studied, and produce a deeper insight on the topic, with fruitful consequences on applications. The aim of this Special Issue is to collect papers that illustrate some directions in which recent research moves, as well as to provide a wide overview on several new approaches to the problem of tensor decomposition.

Prof. Dr. Luca Chiantini
Guest Editor

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Keywords

  • Tensor analysis
  • Rank, border rank and typical rank
  • Complexity
  • Identifiability
  • Secant varieties
  • Segre and Veronese varieties
  • Interpolation problems

Published Papers (5 papers)

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Research

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86 pages, 894 KiB  
Article
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition
by Alessandra Bernardi, Enrico Carlini, Maria Virginia Catalisano, Alessandro Gimigliano and Alessandro Oneto
Mathematics 2018, 6(12), 314; https://doi.org/10.3390/math6120314 - 08 Dec 2018
Cited by 21 | Viewed by 5872
Abstract
We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only [...] Read more.
We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject. Full article
(This article belongs to the Special Issue Decomposability of Tensors)
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21 pages, 352 KiB  
Article
Seeking for the Maximum Symmetric Rank
by Alessandro De Paris
Mathematics 2018, 6(11), 247; https://doi.org/10.3390/math6110247 - 12 Nov 2018
Cited by 1 | Viewed by 2159
Abstract
We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in [...] Read more.
We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in recent times to find new lower and upper bounds. Full article
(This article belongs to the Special Issue Decomposability of Tensors)
13 pages, 320 KiB  
Article
On Comon’s and Strassen’s Conjectures
by Alex Casarotti, Alex Massarenti and Massimiliano Mella
Mathematics 2018, 6(11), 217; https://doi.org/10.3390/math6110217 - 25 Oct 2018
Cited by 6 | Viewed by 2575
Abstract
Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We [...] Read more.
Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon’s conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties. Full article
(This article belongs to the Special Issue Decomposability of Tensors)
9 pages, 265 KiB  
Article
Set Evincing the Ranks with Respect to an Embedded Variety (Symmetric Tensor Rank and Tensor Rank
by Edoardo Ballico
Mathematics 2018, 6(8), 140; https://doi.org/10.3390/math6080140 - 14 Aug 2018
Viewed by 3138
Abstract
Let X ⊂ P r be an integral and non-degenerate variety. We study when a finite set S ⊂ X evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the [...] Read more.
Let X ⊂ P r be an integral and non-degenerate variety. We study when a finite set S ⊂ X evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the order d Veronese embedding X n , d of P n and | S | ≤ ( n + ⌊ d / 2 ⌋ n ) . For the tensor rank, we describe the cases with | S | ≤ 3 . For X n , d , we raise some questions of the maximum rank for d ≫ 0 (for a fixed n) and for n ≫ 0 (for a fixed d). Full article
(This article belongs to the Special Issue Decomposability of Tensors)

Review

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19 pages, 288 KiB  
Review
A Very Brief Introduction to Nonnegative Tensors from the Geometric Viewpoint
by Yang Qi
Mathematics 2018, 6(11), 230; https://doi.org/10.3390/math6110230 - 30 Oct 2018
Cited by 5 | Viewed by 2315
Abstract
This note is a short survey of nonnegative tensors, primarily from the geometric point of view. In addition to basic definitions, we discuss properties of and questions about nonnegative tensors, which may be of interest to geometers. Full article
(This article belongs to the Special Issue Decomposability of Tensors)
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