Tests of Non-Equivalence among Absolutely Nonsingular Tensors through Geometric Invariants

TL;DR

This paper investigates how affine geometric invariants of determinant polynomials can distinguish non-equivalent 4x4x3 absolutely nonsingular tensors, combining theoretical insights with numerical calculations to advance tensor data analysis.

Contribution

It introduces a novel affine geometric approach using invariants of determinant polynomial surfaces to identify non-equivalence among nonsingular tensors.

Findings

Affine geometric invariants effectively discriminate tensor non-equivalence.

Numerical calculations confirm the usefulness of these invariants.

20-spherical design aids in invariant calculation.

Abstract

4x4x3 absolutely nonsingular tensors are characterized by their determinant polynomial. Non-quivalence among absolutely nonsingular tensors with respect to a class of linear transformations, which do not chage the tensor rank,is studied. It is shown theoretically that affine geometric invariants of the constant surface of a determinant polynomial is useful to discriminate non-equivalence among absolutely nonsingular tensors. Also numerical caluculations are presented and these invariants are shown to be useful indeed. For the caluculation of invarinats by 20-spherical design is also commented. We showed that an algebraic problem in tensor data analysis can be attacked by an affine geometric method.