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Low-Rank Tensor Decomposition over Finite Fields

Jason Yang

TL;DR

This work studies rank-$R$ decompositions of a $3$D tensor over finite fields, showing that for fixed $R\le 4$ such decompositions (or proof of nonexistence) can be decided in polynomial time using a slice-basis reduction and rank-factorization invariants. It also proves NP-hardness for rank-2 decompositions when wildcards are allowed over $\mathbb{Z}/2\mathbb{Z}$ via a NAE-3SAT construction, while giving polynomial-time algorithms for rank-1 wildcard decompositions both in 3D and for matrices. The approach combines linear-algebraic tools (row-reduction, GL-actions, full-rank factorizations) with structured case analysis on the reduced forms of the coefficient matrices, enabling control of combinatorial explosion. The results inform the boundary between tractable and intractable tensor decompositions over finite fields and highlight interesting directions for extending tractability to higher ranks and broader number systems with potential implications for explicit constructions in fast matrix multiplication and algebraic complexity.

Abstract

We show that finding rank-$R$ decompositions of a 3D tensor, for $R\le 4$, over a fixed finite field can be done in polynomial time. However, if some cells in the tensor are allowed to have arbitrary values, then rank-2 is NP-hard over the integers modulo 2. We also explore rank-1 decomposition of a 3D tensor and of a matrix where some cells are allowed to have arbitrary values.

Low-Rank Tensor Decomposition over Finite Fields

TL;DR

This work studies rank- decompositions of a D tensor over finite fields, showing that for fixed such decompositions (or proof of nonexistence) can be decided in polynomial time using a slice-basis reduction and rank-factorization invariants. It also proves NP-hardness for rank-2 decompositions when wildcards are allowed over via a NAE-3SAT construction, while giving polynomial-time algorithms for rank-1 wildcard decompositions both in 3D and for matrices. The approach combines linear-algebraic tools (row-reduction, GL-actions, full-rank factorizations) with structured case analysis on the reduced forms of the coefficient matrices, enabling control of combinatorial explosion. The results inform the boundary between tractable and intractable tensor decompositions over finite fields and highlight interesting directions for extending tractability to higher ranks and broader number systems with potential implications for explicit constructions in fast matrix multiplication and algebraic complexity.

Abstract

We show that finding rank- decompositions of a 3D tensor, for , over a fixed finite field can be done in polynomial time. However, if some cells in the tensor are allowed to have arbitrary values, then rank-2 is NP-hard over the integers modulo 2. We also explore rank-1 decomposition of a 3D tensor and of a matrix where some cells are allowed to have arbitrary values.
Paper Structure (15 sections, 9 theorems, 3 equations)

This paper contains 15 sections, 9 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Notation
  4. Without wildcards
  5. Preliminary results about matrix factorizations
  6. Algorithm
  7. Preprocessing
  8. Single row
  9. Rows sharing one common column
  10. Rows sharing two common columns
  11. Runtime
  12. With wildcards
  13. Rank 1 with wildcards
  14. Rank-1 with wildcards for matrices
  15. Future directions

Key Result

Theorem 1

For fixed $R\le 4$, finding a rank-$R$ decomposition over a finite field $\mathbb{F}$ of an $n\times n\times n$ tensor, or determining that it does not exist, can be done in $O(n^3+f(|\mathbb{F}|,R)n^2)$ time, for some function $f$.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Corollary 1
  • ...and 4 more