Related papers: About Notations in Multiway Array Processing
Tensor network diagram (graphical notation) is a useful tool that graphically represents multiplications between multiple tensors using nodes and edges. Using the graphical notation, complex multiplications between tensors can be described…
The Kronecker product is an invaluable tool for data-sparse representations of large networks and matrices with countless applications in machine learning, graph theory and numerical linear algebra. In some instances, the sparsity pattern…
Hypergraphs and tensors extend classic graph and matrix theory to account for multiway relationships, which are ubiquitous in engineering, biological, and social systems. While the Kronecker product is a potent tool for analyzing the…
The Kronecker product is a key algorithm and is ubiquitous across the physical, biological, and computation social sciences. Thus considerations of optimal implementation are important. The need to have high performance and computational…
The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that…
Tensors or {\em multi-way arrays} are functions of three or more indices $(i,j,k,\cdots)$ -- similar to matrices (two-way arrays), which are functions of two indices $(r,c)$ for (row,column). Tensors have a rich history, stretching over…
We give a detailed exposition of the "vectorized" notation for dealing with quantum operations. This notation is used to highlight the relationships between representations of completely-positive dynamics. Vectorization considerably…
Many applications produce multiway data of exceedingly high dimension. Modeling such multi-way data is important in multichannel signal and video processing where sensors produce multi-indexed data, e.g. over spatial, frequency, and…
A nonstandard application of bivariate polynomial interpolation is discussed: the implicitization of a rational algebraic curve given by its parametric equations. Three different approaches using the same interpolation space are considered,…
Modern datasets are often in the form of matrices or arrays,potentially having correlations along each set of data indices. For example, data involving repeated measurements of several variables over time may exhibit temporal correlation as…
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…
Matrix pencils, or pairs of matrices, may be used in a variety of applications. In particular, a pair of matrices (E,A) may be interpreted as the differential equation E x' + A x = 0. Such an equation is invariant by changes of variables,…
Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also…
Integer iteration rules such as n |-> {a n + b, c n +d} are studied as minimal examples of the general process of multicomputation. Despite the simplicity of such rules, their multiway graphs can be complex, exhibiting, for example,…
We study the commutation relations and normal ordering between families of operators on symmetric functions. These operators can be naturally defined by the operations of multiplication, Kronecker product, and their adjoints. As…
In mathematics, many notations have been invented for the concise representation of mathematical formulae. Tensor index notation is one of such notations and has been playing a crucial role in describing formulae in mathematical physics.…
A definition for functions of multidimensional arrays is presented. The definition is valid for third-order tensors in the tensor t-product formalism, which regards third-order tensors as block circulant matrices. The tensor function…
In this paper we consider four basic multidimensional matrix operations (outer product, Kronecker product, contraction, and projection) and two derivative operations (dot and circle products). We start with the interrelations between these…
We present a formal language with expressions denoting general symbol structures and queries which access information in those structures. A sequence-to-sequence network processing this language learns to encode symbol structures and query…
Over the real numbers, the Kronecker sum is the unique operation on matrices which exponentiates to the Kronecker product. Kronecker quotients provide an algebraic view of decompositions of matrices in terms of Kronecker products. This…