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The notation G × H has often been used for Cartesian products of graphs, but is now more commonly used for another construction known as the tensor product of graphs. The square symbol is intended to be an intuitive and unambiguous notation for the Cartesian product, since it shows visually the four edges resulting from the Cartesian product ...
“An Amazon email scam can look exactly like a real Amazon email, or can be poorly crafted, and everything in between,” according to Alex Hamerstone, a director with the security-consulting ...
In machine learning, the term tensor informally refers to two different concepts (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the strict mathematical sense, a tensor is a multilinear mapping over a set of domain vector spaces to a range vector ...
The adjacency matrix of G × H is the Kronecker (tensor) product of the adjacency matrices of G and H. If a graph can be represented as a tensor product, then there may be multiple different representations (tensor products do not satisfy unique factorization) but each representation has the same number of irreducible factors.
A multi-way graph with K perspectives is a collection of K matrices ,..... with dimensions I × J (where I, J are the number of nodes). This collection of matrices is naturally represented as a tensor X of size I × J × K. In order to avoid overloading the term “dimension”, we call an I × J × K tensor a three “mode” tensor, where “modes” are the numbers of indices used to index ...
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis.
Xerus [52] is a C++ tensor algebra library for tensors of arbitrary dimensions and tensor decomposition into general tensor networks (focusing on matrix product states). It offers Einstein notation like syntax and optimizes the contraction order of any network of tensors at runtime so that dimensions need not be fixed at compile-time.
The strong product of any two graphs can be constructed as the union of two other products of the same two graphs, the Cartesian product of graphs and the tensor product of graphs. An example of a strong product is the king's graph, the graph of moves of a chess king on a chessboard, which can be constructed as a strong product of path graphs ...