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Homotopy techniques for tensor decomposition and perfect identifiability

Jonathan D. Hauenstein, Luke Oeding, Giorgio Ottaviani, Andrew J. Sommese

TL;DR

The paper investigates generic identifiability for perfect tensor formats using a blend of numerical algebraic geometry and algebraic techniques. By applying monodromy loops and alpha-certification, the authors enumerate all decompositions of a general tensor when the format is perfect, and prove two new identifiable cases: $(3,4,5)$ with a unique decomposition into 6 summands and $(2,2,2,3)$ with a unique decomposition into 4 summands. They develop and deploy pseudowitness sets and Koszul flattenings in tandem with apolarity to certify counts and recover decompositions, and they prove identifiability in these new cases via vector-bundle arguments and Chern-class calculations. The work culminates in a conjecture that, besides the classical matrix-pencil case, these two formats are the only identifiable perfect formats, with additional proofs provided for the two new cases and plans to extend the methodology further.

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.

Homotopy techniques for tensor decomposition and perfect identifiability

TL;DR

The paper investigates generic identifiability for perfect tensor formats using a blend of numerical algebraic geometry and algebraic techniques. By applying monodromy loops and alpha-certification, the authors enumerate all decompositions of a general tensor when the format is perfect, and prove two new identifiable cases: with a unique decomposition into 6 summands and with a unique decomposition into 4 summands. They develop and deploy pseudowitness sets and Koszul flattenings in tandem with apolarity to certify counts and recover decompositions, and they prove identifiability in these new cases via vector-bundle arguments and Chern-class calculations. The work culminates in a conjecture that, besides the classical matrix-pencil case, these two formats are the only identifiable perfect formats, with additional proofs provided for the two new cases and plans to extend the methodology further.

Abstract

Let T be a general complex tensor of format . When the fraction 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.

Paper Structure

This paper contains 18 sections, 7 theorems, 53 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Some known results on the number of tensor decompositions
  3. General tensors
  4. The symmetric case
  5. Homotopy techniques for tensor decomposition
  6. Decomposition via monodromy loops
  7. The number of decompositions for perfect format tensors
  8. The number of decompositions for symmetric tensors
  9. Pseudowitness sets and verification
  10. Using pseudowitness sets
  11. Tensors of format $(3, 6, 6)$ of rank $8$.
  12. Tensor decomposition via apolarity
  13. A uniform view of (Koszul) flattenings
  14. Apolarity Lemma for Koszul flattenings
  15. The $3\times 4\times 5$ case
  16. ...and 3 more sections

Key Result

Theorem 1.1

A general tensor of format $(3,4,5)$ has a unique decomposition (tensordec:eq) as a sum of $6$ decomposable summands.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4: Galuppi--Mella GM
  • Theorem 2.1
  • proof
  • Theorem 2.2: Alexander-HirschowitzAH
  • Remark 2.3
  • Remark 4.1
  • Proposition 5.1
  • ...and 7 more