Related papers: Tensors in computations
We shall discuss how the notions of multicategories and their linear representations are related with tensor categories. When one focuses on the ones arizing from planar diagrams, it particularly implies that there is a natural one-to-one…
Eigenvectors of tensors, as studied recently in numerical multilinear algebra, correspond to fixed points of self-maps of a projective space. We determine the number of eigenvectors and eigenvalues of a generic tensor, and we show that the…
Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear…
Computer algebra is widely used in various fields of mathematics, physics and other sciences. The simplification of tensor expressions is an important special case of computer algebra. In this paper, we consider the reduction of tensor…
Tensor algebra is essential for data-intensive workloads in various computational domains. Computational scientists face a trade-off between the specialization degree provided by dense tensor algebra and the algorithmic efficiency that…
Mechanistic interpretability aims to break models into meaningful parts; verifying that two such parts implement the same computation is a prerequisite. Existing similarity measures evaluate either empirical behaviour, leaving them blind to…
In recent years several classes of structured matrices are extended to classes of tensors in the context of tensor complementarity problem. The tensor complementarity problem is a class of nonlinear complementarity problem where the…
We propose a novel framework to understand the text by converting sentences or articles into video-like 3-dimensional tensors. Each frame, corresponding to a slice of the tensor, is a word image that is rendered by the word's shape. The…
The paper presents a REDUCE program for the simplification of tensor expressions that are considered as formal indexed objects. The proposed algorithm is based on the consideration of tensor expressions as vectors in some linear space. This…
A tensor space is a vector space equipped with a finite collection of multilinear forms. The length of a tensor space is its length as a representation of its symmetry group. Infinite dimension tensor spaces of finite length are special,…
We define tensors, corresponding to cubic polynomials, which have the same exponent $\omega$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix…
Numerical applications and, more recently, machine learning applications rely on high-dimensional data that is typically organized into multi-dimensional tensors. Many existing frameworks, libraries, and domain-specific languages support…
This work addresses the problem of learning sparse representations of tensor data using structured dictionary learning. It proposes learning a mixture of separable dictionaries to better capture the structure of tensor data by generalizing…
One way to study an hypergraph is to attach to it a tensor. Tensors are a generalization of matrices, and they are an efficient way to encode information in a compact form. In this paper we study how properties of weighted hypergraphs are…
We derive transformation formulas for the generalized polarization tensors under rigid motions and scaling in three dimensions, and use them to construct an infinite number of invariants under those transformations. These invariants can be…
We define new norms for symmetric tensors over ordered normed spaces; these norms are defined by considering linear combinations of tensor products or powers of positive elements only. Relations between the different norms are studied. The…
In the literature various notions of nonlocal curvature can be found. Here we propose a notion of nonlocal curvature tensor. This we do by generalizing an appropriate representation of the classical curvature tensor and by exploiting some…
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 present a fit-for-purpose introduction to tensors and their operations. It is envisaged to help the reader become acquainted with its underpinning concepts for the study of path signatures. The text includes exercises, solutions and many…
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…