Related papers: About Notations in Multiway Array Processing
Compound matrices play an important role in many fields of mathematics and have recently found new applications in systems and control theory. However, the explicit formulas for these compounds are non-trivial and not always easy to use.…
We introduce a new symbolic representation based on an original generalization of counter abstraction. Unlike classical counter abstraction (used in the analysis of parameterized systems with unordered or unstructured topologies) the new…
An algebraic representation of the Turing machines is given, where the configurations of Turing machines are represented by 4 order tensors, and the transition functions by 8 order tensors. Two types of tensor product are defined, one is to…
Graphical tensor notation is a simple way of denoting linear operations on tensors, originating from physics. Modern deep learning consists almost entirely of operations on or between tensors, so easily understanding tensor operations is…
We define a general product of two $n$-dimensional tensors $\mathbb {A}$ and $\mathbb {B}$ with orders $m\ge 2$ and $k\ge 1$, respectively. This product is a generalization of the usual matrix product, and satisfies the associative law.…
Multivariate versions of the Kronecker theorem in the continuous multivariate setting has recently been published. These theorems characterize the symbols that give rise to finite rank multidimensional Hankel and Toeplitz type operators…
We investigate the iterated Kronecker product of a square matrix with itself and prove an invariance property for symmetric subspaces. This motivates the definition of an iterated symmetric Kronecker product and the derivation of an…
Multiway data analysis aims to uncover patterns in data structured as multi-indexed arrays, with multiway covariance playing a crucial role in many applications. However, the high dimensionality of multiway covariance presents significant…
A tensor is a multidimensional array of numbers that can be used to store data, encode a computational relation and represent quantum entanglement. In this sense a tensor can be viewed as valuable resource whose transformation can lead to…
We propose a novel method for the construction of orthogonal arrays. The algorithm makes use of the Kronecker Product operator in association with unit column vectors to generate new orthogonal arrays from existing orthogonal arrays. The…
As the field of recommender systems has developed, authors have used a myriad of notations for describing the mathematical workings of recommendation algorithms. These notations ap-pear in research papers, books, lecture notes, blog posts,…
Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity…
We review different descriptions of many--body quantum systems in terms of tensor product states. We introduce several families of such states in terms of known renormalization procedures, and show that they naturally arise in that context.…
Two successive generalizations of the usual tensor products are given. One can be constructed for arbitrary binary operations, and not only for semigroups, groups or vector spaces. The second one, still more general, is constructed for…
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,…
We review the properties of the Kronecker (direct, or tensor) product of square matrices $A \otimes B \otimes C \cdots$ in terms of Hubbard operators. In its simplest form, a Hubbard operator $X_n^{i,j}$ can be expressed as the $n$-square…
Machine learning and data mining algorithms are becoming increasingly important in analyzing large volume, multi-relational and multi--modal datasets, which are often conveniently represented as multiway arrays or tensors. It is therefore…
Derksen proved that the spectral norm is multiplicative with respect to vertical tensor products (also known as tensor Kronecker products). We will use this result to show that the nuclear norm and other norms of interest are also…
The use of symbolic knowledge representation and reasoning as a way to resolve the lack of transparency of machine learning classifiers is a research area that lately attracts many researchers. In this work, we use knowledge graphs as the…
To determine the optimal set of hyperparameters of a Gaussian process based on a large number of training data, both a linear system and a trace estimation problem must be solved. In this paper, we focus on establishing numerical methods…