Homotopy techniques for tensor decomposition and perfect identifiability

TL;DR

This paper introduces homotopy techniques to compute all tensor decompositions for certain complex tensors, leading to new results on generic identifiability and conjecture of uniqueness in tensor decomposition.

Contribution

It develops a homotopy-based method to find all tensor decompositions, identifying new cases of generic identifiability and providing algebraic geometry proofs for these cases.

Findings

Successfully identified two new cases of generic tensor identifiability.

Homotopy techniques can find all decompositions starting from one.

Predicted and confirmed the uniqueness of tensor decompositions in new formats.

Abstract

Let T be a general complex tensor of format $(n_1,...,n_d)$. When the fraction $\prod_in_i/[1+\sum_i(n_i-1)]$ is an integer, and a natural inequality (called balancedness) is satisfied, it is expected that T has finitely many minimal decomposition as a sum of decomposable tensors. We show how homotopy techniques allow us to find all the decompositions of T, starting from a given one. Computationally, this gives a guess regarding the total number of such decompositions. This guess matches exactly with all cases previously known, and predicts several unknown cases. Some surprising experiments yielded two new cases of generic identifiability: formats (3,4,5) and (2,2,2,3) which have a unique decomposition as the sum of 6 and 4 decomposable tensors, respectively. We conjecture that these two cases together with the classically known matrix pencils are the only cases where generic identifiability holds, i.e., the only identifiable cases. Building on the computational experiments, we use algebraic geometry to prove these two new cases are indeed generically identifiable.