One example of general unidentifiable tensors

TL;DR

This paper demonstrates that certain balanced tensors in complex spaces are not uniquely identifiable, showing that a tensor of rank 8 in C^3⊗C^6⊗C^6 has at least 6 different decompositions, challenging assumptions about tensor identifiability.

Contribution

It provides the first known example of a balanced tensor in C^3⊗C^6⊗C^6 that is not k-identifiable for k less than the generic rank.

Findings

A rank 8 tensor in C^3⊗C^6⊗C^6 has at least 6 decompositions.

This is the highest known example of non-identifiable balanced tensors with dimension 3.

The result impacts the understanding of tensor decompositions and identifiability in algebraic statistics.

Abstract

The identifiability of parameters in a probabilistic model is a crucial notion in statistical inference. We prove that a general tensor of rank 8 in C^3\otimes C^6\otimes C^6 has at least 6 decompositions as sum of simple tensors, so it is not 8-identifiable. This is the highest known example of balanced tensors of dimension 3, which are not k-identifiable, when k is smaller than the generic rank.