Multidimensional Matrix Network Diagrams

Abstract

Visualization is an important type of information representation used to simplify the perception of information. In 1971, Roger Penrose in his article “Applications of negative dimensional tensors” introduced rules for constructing graphical models of convolution tensor operations, thus visualizing them. Convolutions of multidimensional data arrays are widely used in database development tasks, olap-cube construction, CNN, graph-based tasks, image processing tasks and many others. According to the works of Victor Munerman, the multidimensional matrix algebra is preferred to tensor algebra as an algebra for working with multidimensional objects in the framework of the above problems. In this article, the authors analyze Roger's graphical representation rules for their applicability to convolution operations of multidimensional matrix algebra. The analysis reveals that the available rules are not convenient to use for large numbers of convolution indices and that they are not applicable to (λ, μ)-convolution product of multidimensional matrices. The authors give definitions of convolution operations of the algebra of multidimensional matrices, show their specificity, and construct their diagrams. As an example of using these diagrams, the authors give a solution to the problem of constructing all possible routes in an oriented acyclic graph by means of (1, 0)-convolution product. The authors give an example of using the multidimensional matrix network diagram to solve the encryption problem. The obtained results can be used both for teaching students the algebra of multidimensional matrices and for building a graphical interface for interaction with a hardware-software complex realizing the algebra of multidimensional matrices.