Related papers: Demystifying Tubal Tensor Algebra
The tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features like Singular Value Decomposition and Eckart-Young-like optimality results. Underlying the tubal tensor…
The $\star_M$-family of tensor-tensor products is a framework which generalizes many properties from linear algebra to third order tensors. Here, we investigate positive semidefiniteness and semidefinite programming under the…
The tensor-tensor product (t-product) [M. E. Kilmer and C. D. Martin, 2011] is a natural generalization of matrix multiplication. Based on t-product, many operations on matrix can be extended to tensor cases, including tensor SVD, tensor…
We propose a tensor neural network ($t$-NN) framework that offers an exciting new paradigm for designing neural networks with multidimensional (tensor) data. Our network architecture is based on the $t$-product (Kilmer and Martin, 2011), an…
In past few decades, tensor algebra also known as multi-linear algebra has been developed and customized as a tool to be used for various engineering applications. In particular, with the help of a special form of tensor contracted product,…
Since its introduction by Gauss, Matrix Algebra has facilitated understanding of scientific problems, hiding distracting details and finding more elegant and efficient ways of computational solving. Today's largest problems, which often…
Tensor decompositions have become essential tools for feature extraction and compression of multiway data. Recent advances in tensor operators have enabled desirable properties of standard matrix algebra to be retained for multilinear…
We introduce and study the definition, main properties and applications of iterated twisted tensor products of algebras, motivated by the problem of defining a suitable representative for the product of spaces in noncommutative geometry. We…
The purpose of the present paper is to lay the foundations for a systematic study of tensor products of operator systems. After giving an axiomatic definition of tensor products in this category, we examine in detail several particular…
We generalize the tensor product theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a ''conformal vertex algebra'' or even more generally,…
Recent advances in {matrix-mimetic} tensor frameworks have made it possible to preserve linear algebraic properties for multilinear data analysis and, as a result, to obtain optimal representations of multiway data. Matrix mimeticity arises…
In the framework of quantum mechanics over a quadratic extension of the ultrametric field of p-adic numbers, we introduce a notion of tensor product of p-adic Hilbert spaces. To this end, following a standard approach, we first consider the…
We present the basic concepts of tensor products of vector spaces, emphasizing linear algebraic and combinatorial techniques as needed for applied areas of research. The topics include (1) Introduction; (2) Basic multilinear algebra; (3)…
Theoretical studies have proven that the Hilbert space has remarkable performance in many fields of applications. Frames in tensor product of Hilbert spaces were introduced to generalize the inner product to high-order tensors. However,…
In the present paper, we introduce new tensor Krylov subspace methods for solving linear tensor equations. The proposed methods use the well known T-product for tensors and tensor subspaces related to tube fibers. We introduce some new…
This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a…
It is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies. We look at twisted tensor products, and investigate to which extend this is still true. We give an explicit description of the…
A valuable feature of the tubal tensor framework is that many familiar constructions from matrix algebra carry over to tensors, including SVD and notions of rank. Importantly, it has been shown that for a specific family of tubal products,…
We generalize the tensor product theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a ``conformal vertex algebra'' or even more generally,…
Tubal scalars are usual vectors, and tubal matrices are matrices with every element being a tubal scalar. Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be…