Refined methods for the identifiability of tensors

TL;DR

This paper establishes nearly optimal bounds for the uniqueness of tensor decompositions in high-dimensional tensors of sizes 2^n and 3^n, advancing understanding of tensor identifiability.

Contribution

It provides new nearly optimal bounds for the identifiability of general tensors of specific sizes, refining previous results in tensor decomposition theory.

Findings

Unique decomposition for tensors of size 2^n with rank up to 0.9997*(2^n)/(n+1)

Unique decomposition for tensors of size 3^n with rank up to 0.998*(3^n)/(2n+1)

Extended results with weaker bounds for tensors of arbitrary sizes

Abstract

We prove that the general tensor of size 2^n and rank k has a unique decomposition as the sum of decomposable tensors if k<= 0.9997 (2^n)/(n+1) (the constant 1 being the optimal value). Similarly, the general tensor of size 3^n and rank k has a unique decomposition as the sum of decomposable tensors if k<= 0.998 (3^n)/(2n+1) (the constant 1 being the optimal value). Some results of this flavor are obtained for tensors of any size, but the explicit bounds obtained are weaker.