Generalized inverses of Boolean tensors via the Einstein product

Abstract

Applications of the theory and computations of Boolean matrices are of fundamental importance to study a variety of discrete structural models. But the increasing ability of data collection systems to store huge volumes of multidimensional data, the Boolean matrix representation of data analysis is not enough to represent all the information content of the multiway data in different fields. From this perspective, it is appropriate to develop an infrastructure that supports reasoning about the theory and computations. In this paper, we discuss the generalized inverses of the Boolean tensors with the Einstein product. Further, we elaborate on this theory by producing a few characterizations of different generalized inverses and several equivalence results on Boolean tensors. We explore the space decomposition of the Boolean tensors and present reflexive generalized inverses through it. In addition to this, we address rank and the weight for the Boolean tensor. generalizations of matrices [21,35]. Here the notion of tensors is different in physics and engineering (such as stress tensors) [31], which are generally referred to as tensor fields in mathematics [12]. However, it will be more appropriate if we study the Boolean tensors and the generalized inverses of Boolean tensors. Hence the generalized inverses of Boolean tensors will encounter in many branches of mathematics, including relations theory [34], logic, graph theory, lattice theory [8] and algebraic semigroup theory.Recently, there has been increasing interest in studying inverses [9] and different generalized inverses of tensors based on the Einstein product [5,17,40,41], and opened new perspectives for solving multilinear systems [21,26]. In [17,40], the authors have introduced some basic properties of the range and null space of multidimensional arrays. Further, in [40], it was discussed the adequate definition of the tensor rank, termed as reshaping rank. Corresponding representations of the weighted Moore-Penrose inverse introduced in [4,16] and investigated a few characterizations in [33]. Though this work is focusing on the binary case; i.e., concentrating some interesting results based on the Boolean tensors and generalized inverses of Boolean tensors via the Einstein product. In many instances, the result in the general case does not immediately follow even though it is not difficult to conclude.On the other hand, one of the most successful developments in the world of multilinear algebra is the concept of tensor decomposition [20,21,23]. This concept gives a clear and convenient way to implement all basic operations efficiently. Recently this concept is extended in Boolean tensors [14,18,38]. Further, the fast and scalable distributed algorithms for Boolean tensor decompositions were discussed in [29]. In addition to that, a few applications of these decompositions are discussed in [14,28] for information extraction and clustering. At that same time, Brazell, et al. in [9] discussed decomposition of tensors from the isomorphic group structure on...

Computation of outer inverses of tensors using the QR decomposition

“…The following results discussed in the framework of Boolean tensors in Behera and Sahoo (2020). However, these two results are true for an arbitrary-order tensor.…”

“…The group and Drazin inverse of A are labeled by A # and A D , respectively. Some preliminary results about the tensor range and null space can be collected from (Behera and Sahoo 2020;Ji and Wei 2018;Stanimirović et al 2018).…”