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An efficient uniqueness theorem for overcomplete tensor decomposition

Pascal Koiran

TL;DR

This work provides a constructive uniqueness theorem for order-3 tensors of format $n\times n\times p$ with $p\ge4$, establishing essential uniqueness of rank-$r$ decompositions for generic tensors in the regime $n\le r\le 4n/3$. The authors introduce and exploit the method of commuting extensions, building on Strassen’s lower bounds, to derive an algorithmic decomposition that runs in polynomial time in the input size (and also in the bit-model for rational inputs). They show how to transform the tensor rank problem into a commuting-extension problem, solve it via a companion algorithm, and then apply simultaneous diagonalization to extract the rank-one terms, yielding the first efficient algorithms for overcomplete decomposition of generic order-3 tensors in this setting. The paper also proves NP-hardness for computing commuting extensions in general, places the results in the context of Jennrich/Kruskal-type uniqueness theorems, and discusses recent follow-up work improving rank bounds. Overall, the approach provides a practical, theoretically grounded path to recovering unique decompositions in the overcomplete regime for generic tensors.

Abstract

We give a new, constructive uniqueness theorem for tensor decomposition. It applies to order 3 tensors of format $n \times n \times p$ and can prove uniqueness of decomposition for generic tensors up to rank $r=4n/3$ as soon as $p \geq 4$. One major advantage over Kruskal's uniqueness theorem is that our theorem has an algorithmic proof, and the resulting algorithm is efficient. Like the uniqueness theorem, it applies in the range $n \leq r \leq 4n/3$. As a result, we obtain the first efficient algorithm for overcomplete decomposition of generic tensors of order 3. For instance, prior to this work it was not known how to efficiently decompose generic tensors of format $n \times n \times n$ and rank $r=1.01n$ (or rank $r \leq (1+ε) n$, for some constant $ε>0$). Efficient overcomplete decomposition of generic tensors of format $n \times n \times 3$ remains an open problem. Our results are based on the method of commuting extensions pioneered by Strassen for the proof of his $3n/2$ lower bound on tensor rank and border rank. In particular, we rely on an algorithm for the computation of commuting extensions recently proposed in a companion paper, and on the classical diagonalization-based "Jennrich algorithm" for undercomplete tensor decomposition. This is an updated version of a paper presented at SODA 2025. As a new result, we answer a question from that paper by giving a NP-hardness result for the computation of commuting extensions. The proof relies on a recent construction by Shitov. After the paper appearing in the SODA proceedings was written, another algorithm for the overcomplete decomposition of generic tensors of order~3 was proposed by Kothari, Moitra and Wein.

An efficient uniqueness theorem for overcomplete tensor decomposition

TL;DR

This work provides a constructive uniqueness theorem for order-3 tensors of format with , establishing essential uniqueness of rank- decompositions for generic tensors in the regime . The authors introduce and exploit the method of commuting extensions, building on Strassen’s lower bounds, to derive an algorithmic decomposition that runs in polynomial time in the input size (and also in the bit-model for rational inputs). They show how to transform the tensor rank problem into a commuting-extension problem, solve it via a companion algorithm, and then apply simultaneous diagonalization to extract the rank-one terms, yielding the first efficient algorithms for overcomplete decomposition of generic order-3 tensors in this setting. The paper also proves NP-hardness for computing commuting extensions in general, places the results in the context of Jennrich/Kruskal-type uniqueness theorems, and discusses recent follow-up work improving rank bounds. Overall, the approach provides a practical, theoretically grounded path to recovering unique decompositions in the overcomplete regime for generic tensors.

Abstract

We give a new, constructive uniqueness theorem for tensor decomposition. It applies to order 3 tensors of format and can prove uniqueness of decomposition for generic tensors up to rank as soon as . One major advantage over Kruskal's uniqueness theorem is that our theorem has an algorithmic proof, and the resulting algorithm is efficient. Like the uniqueness theorem, it applies in the range . As a result, we obtain the first efficient algorithm for overcomplete decomposition of generic tensors of order 3. For instance, prior to this work it was not known how to efficiently decompose generic tensors of format and rank (or rank , for some constant ). Efficient overcomplete decomposition of generic tensors of format remains an open problem. Our results are based on the method of commuting extensions pioneered by Strassen for the proof of his lower bound on tensor rank and border rank. In particular, we rely on an algorithm for the computation of commuting extensions recently proposed in a companion paper, and on the classical diagonalization-based "Jennrich algorithm" for undercomplete tensor decomposition. This is an updated version of a paper presented at SODA 2025. As a new result, we answer a question from that paper by giving a NP-hardness result for the computation of commuting extensions. The proof relies on a recent construction by Shitov. After the paper appearing in the SODA proceedings was written, another algorithm for the overcomplete decomposition of generic tensors of order~3 was proposed by Kothari, Moitra and Wein.
Paper Structure (25 sections, 36 theorems, 37 equations, 3 algorithms)

This paper contains 25 sections, 36 theorems, 37 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Uniqueness theorem
  3. Decomposition algorithm.
  4. Our approach: the method of commuting extensions.
  5. Complexity of decomposition algorithms.
  6. Worst case complexity.
  7. Organization of the paper.
  8. Preliminaries
  9. A characterization of tensor rank
  10. The case $r=n$.
  11. General case.
  12. A variation on Strassen's lower bound.
  13. Uniqueness theorems
  14. Uniqueness of commuting extensions.
  15. The new uniqueness theorem.
  16. ...and 10 more sections

Key Result

Theorem 1

Consider a tensor $T=\sum_{i=1}^r u_i \otimes v_i \otimes w_i$ of format $n \times n \times p$ with $p\geq 4$. We assume that $T$ satisfies the following properties: Then $\mathrm{rank}(T)=r$, and the decomposition of $T$ as a sum of $r$ rank one tensors is essentially unique.

Theorems & Definitions (62)

  • Theorem 1: simple form of the uniqueness theorem
  • Theorem 2: Strassen strassen83
  • Definition 3
  • Theorem 4
  • Proposition 5
  • Corollary 6
  • proof
  • Corollary 7
  • proof
  • Corollary 8
  • ...and 52 more