Related papers: The span of singular tuples of a tensor beyond the…

In this paper, we study tensor spaces beyond the boundary format and analyze whether the general critical space coincides with the general span of singular vector tuples. For all tensor spaces exceeding the boundary format by one in an…

In this paper we discuss the notion of singular vector tuples of a complex valued $d$-mode tensor of dimension m_1 x ... x m_d. We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the…

In this paper we study the locus of singular tuples of a complex valued multisymmetric tensor. The main problem that we focus on is: given the set of singular tuples of some general tensor, which are all the tensors that admit those same…

Tree tensor networks such as the tensor train format are a common tool for high dimensional problems. The associated multivariate rank and accordant tuples of singular values are based on different matricizations of the same tensor. While…

Third-order tensors are widely used as a mathematical tool for modeling physical properties of media in solid state physics. In most cases, they arise as constitutive tensors of proportionality between basic physics quantities. The…

Given a tensor $f$ in a Euclidean tensor space, we are interested in the critical points of the distance function from $f$ to the set of tensors of rank at most $k$, which we call the critical rank-at-most-$k$ tensors for $f$. When $f$ is a…

Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a…

A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a sum of symmetric outer product of vectors. A rank-1 order-k…

We introduce an inductive method for the study of the uniqueness of decompositions of tensors, by means of tensors of rank 1. The method is based on the geometric notion of weak defectivity. For three-dimensional tensors of type (a, b, c),…

Based upon the T-SVD (tensor SVD) of third order tensors, introduced by Kilmer and her collaborators, we define T-singular values of third order tensors. T-singular values of third order tensors are nonnegative scalars. The number of…

We give a new, constructive uniqueness theorem for tensor decomposition. It applies to order 3 tensors of format $n \times n \times p$ and can prove uniqueness of decomposition for generic tensors up to rank $r=4n/3$ as soon as $p \geq 4$.…

One of the key problems in tensor completion is the number of uniformly random sample entries required for recovery guarantee. The main aim of this paper is to study $n_1 \times n_2 \times n_3$ third-order tensor completion based on…

Kruskal's theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. In this work, we propose a conjecture in which the k-rank condition of…

We develop deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors, in a spirit similar to classical results for matrices such as those due to Weyl, Davis, Kahan and Wedin. Our bounds…

An $n \times n \times p$ tensor is called a T-square tensor. It arises from many applications, such as the image feature extraction problem and the multi-view clustering problem. We may symmetrize a T-square tensor to a T-symmetric tensor.…

We show that the spectral norm of a random $n_1\times n_2\times \cdots \times n_K$ tensor (or higher-order array) scales as $O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right)$ under some sub-Gaussian assumption on the entries. The proof is…

We show that a generic tensor $T\in \mathbb{F}^{n\times n\times \dots\times n}$ of order $k$ and CP rank $d$ can be uniquely recovered from $n\log n+dn\log \log n +o(n\log \log n) $ uniformly random entries with high probability if $d$ and…

Finding the rank of a tensor is a problem that has many applications. Unfortunately it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of…

We develop a systematic way to solve linear equations involving tensors of arbitrary rank. We start off with the case of a rank $3$ tensor, which appears in many applications, and after finding the condition for a unique solution we derive…

The structure tensor of $\mathfrak{sl}_n$, denoted $T_{\mathfrak{sl}_n}$, is the tensor arising from the Lie bracket bilinear operation on the set of traceless $n \times n$ matrices over $\mathbb{C}$. This tensor is intimately related to…