Decomposability of Tensors Edited by Luca Chiantini Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Decomposability of Tensors Decomposability of Tensors Special Issue Editor Luca Chiantini MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editor Luca Chiantini Universita` degli Studi di Siena Italy Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) in 2018 (available at: https://www.mdpi.com/journal/mathematics/ special issues/Mathematics Tensor Decomposition) For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03897-590-8 (Pbk) ISBN 978-3-03897-591-5 (PDF) c 2019 by the authors. 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Contents About the Special Issue Editor ...................................... vii Preface to ”Decomposability of Tensors” ................................ ix Yang Qi A Very Brief Introduction to Nonnegative Tensors from the Geometric Viewpoint Reprinted from: Mathematics 2018, 6, 230, doi:10.3390/math6110230 ................. 1 Edoardo Ballico Set Evincing the Ranks with Respect to an Embedded Variety (Symmetric Tensor Rank and Tensor Rank Reprinted from: Mathematics 2018, 6, 140, doi:10.3390/math6080140 ................. 20 Alex Casarotti, Alex Massarenti and Massimiliano Mella On Comon’s and Strassen’s Conjectures Reprinted from: Mathematics 2018, 6, 217, doi:10.3390/math6110217 ................. 29 Alessandro De Paris Seeking for the Maximum Symmetric Rank Reprinted from: Mathematics 2018, 6, 247, doi:10.3390/math6110247 ................. 42 Alessandra Bernardi, Enrico Carlini, Maria Virginia Catalisano, Alessandro Gimigliano and Alessandro Oneto The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition Reprinted from: Mathematics 2018, 6, 314, doi:10.3390/math6120314 ................. 63 v About the Special Issue Editor Luca Chiantini was born in 1957 and received his degree in Mathematics in 1979. He is Full Professor of Geometry at the University of Siena, Department of Information Engineering and Mathematical Sciences. He has published more than 90 research papers in Algebraic Geometry and Commutative Algebra, and he has edited books and conference proceedings on these topics. He has been a member of the scientific committee of several international meetings, including the joint meeting of the Unione Matematica Italiana and the real Sociedad Matematica Espanola. He participated in the editorial board of several Italian mathematical journals. Actually, he is a member of the editorial board of the Bolletino dell’Unione Matematica Italiana, and he is a member of the panel of the Scuola Matematica Interuniversitaria. His recent research interests focus on the geometry of special projective varieties, like Veronese and Segre varieties, and their applications in tensor analysis and algebraic statistics. vii Preface to ”Decomposability of Tensors” The Special Issue “tensor decomposition” is devoted to collecting papers on a subject that is rapidly developing in recent years, with (unexpected) connections between different areas of mathematics. Though tensor analysis is a topic that, for a long time, has been considered a chapter of multilinear algebra with a view towards numerical analysis, it turned out recently that methods of projective geometry, often arising from a classical point of view, have a strong connection with the theory of tensors and can produce advances that are also valuable in applicative domains. In particular, the study of tensor rank, i.e., the complexity of a tensor, was invigorated by the introduction of techniques based on the background of projective geometry. If one considers elementary tensors as a tensor product of vectors, then the computation of the rank corresponds to finding the minimum k, such that a tensor belongs to the k-secant variety of a product variety. In projective terms, the main notions of tensor analysis can be defined modulo scalar products, so most problems can be translated in terms of points in projective spaces, and product varieties become Segre embeddings of products of projective spaces (or Veronese embeddings of projective spaces, in the symmetric setting). In this circle of ideas, a decomposition of a tensor T corresponds to a set Z of projective points, such that T belongs to the linear span of Z. Thus, the geometry of secant varieties to Segre and Veronese varieties provides basic tools for understanding the decomposition of a given tensor. Notions like the uniqueness and minimality of a given decomposition found a natural formulation in terms of projective geometry. Also, special decompositions can be described in terms of proprieties of secant spaces. Altogether, the initial features of these new perspectives were described in a series of books and papers. Yet, as the theory develops quickly, it is useful to make frequent reports on the status of the art. This book collects papers that contain both surveys on the actual main achievements on some classical problems on the decomposition of tensors, like the best-known bounds on the rank of symmetric tensors, together with results on special decompositions and extensions to the generalized study of decompositions with respect to any subvarieties. We hope that the content of this book will provide a helpful collection of geometric perspectives on tensor analysis and tensor decomposition, which are necessary both to create a solid starting point for future developments and to establish a background of geometric methods for people who arrived to work in the subject coming from different points of view. Luca Chiantini Special Issue Editor ix mathematics Review A Very Brief Introduction to Nonnegative Tensors from the Geometric Viewpoint Yang Qi Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA; [email protected] Received: 27 September 2018; Accepted: 24 October 2018; Published: 30 October 2018 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. Keywords: nonnegative tensors; low-rank approximations; uniqueness and identifiability; spectral theory; EM algorithm; semialgebraic geometry 1. Introduction Tensors are ubiquitous in mathematics and sciences. In the study of complex and real tensors, algebraic geometry has demonstrated its power [1,2]. On the other hand, tensor computations also help people understand classical algebraic varieties, such as the secant varieties of Segre varieties and Veronese varieties, and raise interesting and challenging questions in algebraic geometry [3,4]. Traditionally, geometers tend to study tensors in a coordinate-free way. However, in applications, practitioners must work with coordinates. Among those tensors widely used in practice, a large number of them are nonnegative tensors, i.e., tensors with nonnegative entries. In this case, most powerful geometric tools developed for complex tensors can not be applied directly due to the fact that the Euclidean closure of tensors with rank no greater than a fixed integer is no longer a variety, but a semialgebraic set. This forces us to investigate the semialgebraic geometry of nonnegative tensors. In this note, we will review some important properties of nonnegative tensors obtained by studying the semialgebraic geometry, and propose several open problems which are pivotal in understanding nonnegative tensors and also may be interesting to geometers. 2. Definitions Nonnegative tensors arise naturally in many areas, such as hyperspectral imaging, statistics, spectroscopy, computer vision, phylogenetics, and so on. See [5–8] and the references therein. Before further investigations, let us recall basic definitions of tensors. K V = ⊗···⊗ Definition 1. Let V1, ..., Vd be vector spaces over a field . The tensor product V1 Vd is the free ×···× linear space spanned by V1 Vd quotient by the equivalence relation: ( α + β ) ∼ α( )+β( ) v1,..., vi vi,...,vd v1,...,vi,...,vd v1,...,vi,...,vd (1) ∈ α β ∈ K = ⊗···⊗ for every vi, vi Vi, i, i , and i 1, . , d. An element of V1 Vd is called a tensor. ⊗···⊗ Equivalently, V1 Vd is the vector space of multilinear functions: ∗ ×···× ∗ → K V1 Vd , Mathematics 2018, 6, 230; doi:10.3390/math61102301 www.mdpi.com/journal/mathematics Mathematics 2018, 6, 230 ∗ = ( ) where Vi is the dual space of Vi for i 1, ..., d. A representative of the equivalence class of v1, ..., vd ⊗···⊗ is called a decomposable tensor and denoted by v1 vd. ∈ ⊗···⊗ The rank of a given tensor T V1 Vd is the minimum integer r such that T is a sum of r decomposable tensors, i.e., r = ⊗···⊗ T ∑ v1,i vd,i, (2) i=1 ∈ = = where vj,i Vj for j 1, ..., d and i 1, ..., r. Such a decomposition is called a rank decomposition (or canonical polyadic decomposition or CP decomposition). K = R Now we focus on the case , and for each Vi we fix a basis, which enables us to work with coordinates. Let R+ be the semiring of nonnegative real numbers. A nonnegative tensor in ⊗···⊗ + V1 Vd is a tensor whose coordinates are nonnegative. Let Vi denote the set of nonnegative = V+ V vectors in Vi for each i 1, . , d, and denote the set of nonnegative tensors in . + Definition 2. For T ∈ V , the nonnegative rank of T is the minimum integer r so that there exist nonnegative ∈ + = = vectors vi,j Vi for i 1, . , d and j 1, . , r making Equation (2) holds. + + It is clear that rank+(T) ≥ rank(T) for every T ∈ V .