Special Issue "Decomposability of Tensors"

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

Deadline for manuscript submissions: closed (28 September 2018)

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

Special Issue Editor

Guest Editor
Prof. Dr. Luca Chiantini

Dipartimento di Scienze Matematiche e Informatiche, Università degli Studi di Siena, Pian dei Mantellini, 44, 53100 Siena, Italy
Website | E-Mail
Interests: Algebraic Geometry; Projective Geometry; Multilinar Algebra; Commutative Algebra; Computer Algebra; Algebraic Statistics

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

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 850 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

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

Published Papers (5 papers)

View options order results:
result details:
Displaying articles 1-5
Export citation of selected articles as:

Research

Jump to: Review

Open AccessArticle
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition
Mathematics 2018, 6(12), 314; https://doi.org/10.3390/math6120314
Received: 9 October 2018 / Revised: 13 November 2018 / Accepted: 14 November 2018 / Published: 8 December 2018
Cited by 1 | PDF Full-text (894 KB) | HTML Full-text | XML Full-text
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) Printed Edition available
Figures

Figure 1

Open AccessArticle
Seeking for the Maximum Symmetric Rank
Mathematics 2018, 6(11), 247; https://doi.org/10.3390/math6110247
Received: 9 October 2018 / Revised: 2 November 2018 / Accepted: 3 November 2018 / Published: 12 November 2018
PDF Full-text (352 KB) | HTML Full-text | XML Full-text
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) Printed Edition available
Open AccessArticle
On Comon’s and Strassen’s Conjectures
Mathematics 2018, 6(11), 217; https://doi.org/10.3390/math6110217
Received: 18 September 2018 / Revised: 20 October 2018 / Accepted: 22 October 2018 / Published: 25 October 2018
PDF Full-text (320 KB) | HTML Full-text | XML Full-text
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) Printed Edition available
Open AccessArticle
Set Evincing the Ranks with Respect to an Embedded Variety (Symmetric Tensor Rank and Tensor Rank
Mathematics 2018, 6(8), 140; https://doi.org/10.3390/math6080140
Received: 11 July 2018 / Revised: 7 August 2018 / Accepted: 8 August 2018 / Published: 14 August 2018
PDF Full-text (265 KB) | HTML Full-text | XML Full-text
Abstract
Let XPr be an integral and non-degenerate variety. We study when a finite set SX 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) Printed Edition available

Review

Jump to: Research

Open AccessReview
A Very Brief Introduction to Nonnegative Tensors from the Geometric Viewpoint
Mathematics 2018, 6(11), 230; https://doi.org/10.3390/math6110230
Received: 27 September 2018 / Revised: 24 October 2018 / Accepted: 24 October 2018 / Published: 30 October 2018
PDF Full-text (288 KB) | HTML Full-text | XML Full-text
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) Printed Edition available
Mathematics EISSN 2227-7390 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top