Tensor Formulation of the General Linear Model with Einstein Notation

TL;DR

This paper proposes a tensor-based reformulation of the general linear model using Einstein notation, aiming to improve computational efficiency and organizational clarity over traditional matrix methods.

Contribution

It introduces a novel tensor formulation of the general linear model, addressing inefficiencies and inconsistencies in conventional matrix-based approaches.

Findings

Tensor formulation reduces computation time.

Enhanced memory efficiency with multidimensional arrays.

Improved organizational clarity in model representation.

Abstract

The general linear model is a universally accepted method to conduct and test multiple linear regression models. Using this model one has the ability to simultaneously regress covariates among different groups of data. Moreover, there are hundreds of applications and statistical tests associated with the general linear model. However, the conventional matrix formulation is relatively inelegant which yields multiple difficulties including slow computation speed due to a large number of computations, increased memory usage due to needlessly large data structures, and organizational inconsistency. This is due to the fundamental incongruence between the degrees of freedom of the information the data structures in the conventional formulation of the general linear model are intended to represent and the rank of the data structures themselves. Here, I briefly suggest an elegant reformulation of the general linear model which involves the use of tensors and multidimensional arrays as opposed to exclusively flat structures in the conventional formulation. To demonstrate the efficacy of this approach I translate a few common applications of the general linear model from the conventional formulation to the tensor formulation.