Related papers: Triangular tensors and set-intersection problems
We develop the theory of costratification in the setting of relative tensor-triangular geometry, in the sense of Stevenson, providing a unified approach to classification results of Neeman and Benson--Iyengar--Krause, while laying the…
This educational article highlights the geometric and algebraic complexities that distinguish tensors from matrices, to supplement coverage in advanced courses on linear algebra, matrix analysis, and tensor decompositions. Using the case of…
We present a new algorithm for recovering paths from their third-order signature tensors, an inverse problem in rough analysis. Our algorithm provides the exact solution to this learning problem and improves upon current approaches by an…
We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation.…
The theory of matrix splitting is a useful tool for finding solution of rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit theory of weak regular splittings for rectangular…
Random tensor models for a generic complex tensor generalize matrix models in arbitrary dimensions and yield a theory of random geometries. They support a 1/N expansion dominated by graphs of spherical topology. Their Schwinger Dyson…
We study a composition operation on monads, equivalently presented as large equational theories. Specifically, we discuss the existence of tensors, which are combinations of theories that impose mutual commutation of the operations from the…
We introduce the Bhattacharya-Mesner rank of third order hypermatrices as a relaxation to the tensor rank and devise from it some bounds for the tensor rank. We use the Bhattacharya-Mesner rank to extend to third order hypermatrices the…
We define a triangular change of basis in which the form is diagonal and explicitly compute the diagonal entries of this matrix as products of quotients of Chebyshev polynomials, corroborating the determinant computation of Ko and…
Higher-order tensors appear in various areas of mechanics as well as physics, medicine or earth sciences. As these tensors are highly complex, most are not well understood. Thus, the analysis and the visualization process form a highly…
A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming…
We introduce subspace rank as a tool for studying ranks of tensors and X-rank more generally. We derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in C^2 \otimes C^b \otimes C^b. We…
The first part of these lecture notes is mostly devoted to a comparative discussion of the three basic large $N$ limits, which apply to fields which are vectors, matrices, or tensors of rank three and higher. After a brief review of some…
The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we mainly study low rank third-order tensor completion problems by using Riemannian optimization methods on…
In arXiv:2204.03190, we proposed a universal method to reduce one-loop integrals with both tensor structure and higher-power propagators. But the method is quite redundant as it does not utilize the results of lower rank cases when…
The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or…
Due to the explosive growth of large-scale data sets, tensors have been a vital tool to analyze and process high-dimensional data. Different from the matrix case, tensor decomposition has been defined in various formats, which can be…
This is a survey primarily about determining the border rank of tensors, especially those relevant for the study of the complexity of matrix multiplication. This is a subject that on the one hand is of great significance in theoretical…
To determine whether an $n\times n$-matrix has rank at most $r$ it suffices to check that the $(r+1)\times (r+1)$-minors have rank at most $r$. In other words, to describe the set of $n\times n$-matrices with the property of having rank at…
Tensors are often studied by introducing preorders such as restriction and degeneration: the former describes transformations of the tensors by local linear maps on its tensor factors; the latter describes transformations where the local…