Block Tensor Unfoldings

TL;DR

This paper develops a concise notation and theoretical foundation for block tensor computations, including identities and the concept of block tensor unfolding, to support advanced numerical multilinear algebra algorithms.

Contribution

It introduces a new notation, proves key identities, and formalizes the notion of block tensor unfolding for improved tensor computation analysis.

Findings

Established concise notation for block tensors

Proved several useful block tensor identities

Formalized the concept of block tensor unfolding

Abstract

Within the field of numerical multilinear algebra, block tensors are increasingly important. Accordingly, it is appropriate to develop an infrastructure that supports reasoning about block tensor computation. In this paper we establish concise notation that is suitable for the analysis and development of block tensor algorithms, prove several useful block tensor identities, and make precise the notion of a block tensor unfolding.