# A low-rank approximation of tensors and the topological group structure of invertible matrices

## TL;DR

This paper explores the properties of tensor rank, the topological group structure of invertible matrices, and demonstrates the non-existence of best low-rank approximations for certain tensors, using matrix spectral approximations.

## Contribution

It introduces a new perspective on tensor rank and low-rank approximation, linking tensor properties with the topological group structure of invertible matrices.

## Key findings

- Determines that a best rank-n approximation does not exist for certain tensors.
- Connects tensor approximation problems with the spectral properties of matrices.
- Provides insights into the structure of multilinear matrix multiplication.

## Abstract

By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is concerned with properties of the~tensor rank that is a natural generalization of the matrix rank. The topological group structure of invertible matrices is involved in this study. The multilinear matrix multiplication is discussed from a viewpoint of transformation groups. We treat a low-rank tensor approximation in finite-dimensional tensor products. It is shown that the problem on determining a best rank-$n$ approximation for a tensor of size $n\times n \times 2$ has no a solution.To this end, we make use of an~approximation by matrices with simple spectra.