Related papers: Toward a generalization of Kruskal's theorem on te…

Tensor decompositions have rich applications in statistics and machine learning, and developing efficient, accurate algorithms for the problem has received much attention recently. Here, we present a new method built on Kruskal's uniqueness…

The tensor rank decomposition is a useful tool for the geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able…

We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We…

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 prove the existence of an open set of $n_1\times n_2 \times n_3$ tensors of rank $r$ on which a popular and efficient class of algorithms for computing tensor rank decompositions based on a reduction to a linear matrix pencil, typically…

The main goal of this paper is to extend the so-called Dirac-Frenkel Variational Principle in the framework of tensor Banach spaces. To this end we observe that a tensor product of normed spaces can be described as a union of disjoint…

The uniform Kruskal theorem extends the original result for trees to general recursive data types. As shown by A. Freund, M. Rathjen and A. Weiermann, it is equivalent to $\Pi^1_1$-comprehension, over $\mathsf{RCA_0}$ with the chain…

We present novel algorithmic techniques to efficiently verify the Kruskal rank of matrices that arise in sparse linear regression, tensor decomposition, and latent variable models. Our unified framework combines randomized hashing…

In this paper, we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order low rank tensors in a balanced way. We call such a decomposition the triple decomposition, and the…

Rank-metric codes are subspaces of matrices over finite fields endowed with the rank metric and admit a natural tensorial representation. The tensor rank provides a measure of the minimal size of a decomposition of a code into rank-one…

A tensor defined over a finite field $\mathbb{F}$ has low analytic rank if the distribution of its values differs significantly from the uniform distribution. An order $d$ tensor has partition rank 1 if it can be written as a product of two…

We propose a constructive algorithm that decomposes an arbitrary real tensor into a finite sum of orthonormal rank-1 outer products. The algorithm, named TTr1SVD, works by converting the tensor into a tensor-train rank-1 (TTr1) series via…

We show how methods of algebraic geometry can produce criteria for the identifiability of specific tensors that reach beyond the range of applicability of the celebrated Kruskal criterion. More specifically, we deal with the symmetric…

Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan-Hochster in their proof of Stillman's conjecture and generalised here to other tensors, is…

A tensor is a multi-way array that can represent, in addition to a data set, the expression of a joint law or a multivariate function. As such it contains the description of the interactions between the variables corresponding to each of…

Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to $n^{\lfloor p/2 \rfloor}$ for a $p$-th order tensor in…

We investigate the uniqueness of decomposition of general tensors $T\in {\mathbb C}^{n_1+1}\otimes\cdots\otimes{\mathbb C}^{n_r+1}$ as a sum of tensors of rank $1$. This is done extending the theory developed in a previous paper by the…

We show that sufficiently low tensor rank for the balanced tripartitioning tensor $P_d(x,y,z)=\sum_{A,B,C\in\binom{[3d]}{d}:A\cup B\cup C=[3d]}x_Ay_Bz_C$ for a large enough constant $d$ implies uniform arithmetic circuits for the matrix…

Canonical Polyadic Decomposition (CPD) of a third-order tensor is decomposition in a minimal number of rank-$1$ tensors. We call an algorithm algebraic if it is guaranteed to find the decomposition when it is exact and if it only relies on…

We give reconstruction algorithms for subclasses of depth-3 arithmetic circuits. In particular, we obtain the first efficient algorithm for finding tensor rank, and an optimal tensor decomposition as a sum of rank-one tensors, when given…