On generic identifiability of 3-tensors of small rank

TL;DR

This paper presents a new inductive geometric method to determine the uniqueness of tensor decompositions, significantly expanding the known range of ranks for which 3-tensors are identifiable, with applications to small-sized tensors.

Contribution

Introduces a novel inductive approach based on weak defectivity for establishing tensor identifiability, improving the known bounds for 3-tensors of small size.

Findings

Proves k-identifiability for general tensors when k e (a+1)(b+1)/16.

Provides a complete list of small tensors where identifiability fails.

Identifies the 4x4x4 tensor of rank 6 as a notable non-identifiable case.

Abstract

We introduce an inductive method for the study of the uniqueness of decompositions of tensors, by means of tensors of rank 1. The method is based on the geometric notion of weak defectivity. For three-dimensional tensors of type (a, b, c), a\le b\le c, our method proves that the decomposition is unique (i.e. k-identifiability holds) for general tensors of rank k, as soon as k\le (a+1)(b+1)/16. This improves considerably the known range for identifiability. The method applies also to tensor of higher dimension. For tensors of small size, we give a complete list of situations where identifiability does not hold. Among them, there are 4\times4\times4 tensors of rank 6, an interesting case because of its connection with the study of DNA strings.