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Fixed-parameter tractability of canonical polyadic decomposition over finite fields

Jason Yang

TL;DR

This paper shows that computing a rank-$R$ canonical polyadic decomposition of a 3D tensor over a finite field $\mathbb{F}$ is fixed-parameter tractable with respect to $R$ and $|\mathbb{F}|$, and provides a nontrivial upper bound on the running time. The key idea is to extract a basis via row-reduction to reduce to a small core tensor of size $R_0\times R_1\times R_2$ with $R_i\le R$, and then solve the CPD on this core through controlled brute-force search and basis-conversion back to the original tensor. The authors tighten the time bounds with preprocessing and postprocessing that reduce dependence on the ambient dimensions and introduce a CPD solving strategy that leverages fixing a factor and solving a structured linear system, augmented by a greedy scheme to maximize monomial columns. These results establish exact, guaranteed CPDs over finite fields and illuminate the landscape of tensor Decompositions from a fixed-parameter perspective, with implications for exact algorithms in algebraic settings and potential extensions to broader domains like integers. The work also opens questions about optimal constants for fixed parameters and the scope of FPT results beyond finite fields.

Abstract

We present a simple proof that finding a rank-$R$ canonical polyadic decomposition of a 3-dimensional tensor over a finite field $\mathbb{F}$ is fixed-parameter tractable with respect to $R$ and $\mathbb{F}$. We also show a nontrivial upper bound on the time complexity of this problem.

Fixed-parameter tractability of canonical polyadic decomposition over finite fields

TL;DR

This paper shows that computing a rank- canonical polyadic decomposition of a 3D tensor over a finite field is fixed-parameter tractable with respect to and , and provides a nontrivial upper bound on the running time. The key idea is to extract a basis via row-reduction to reduce to a small core tensor of size with , and then solve the CPD on this core through controlled brute-force search and basis-conversion back to the original tensor. The authors tighten the time bounds with preprocessing and postprocessing that reduce dependence on the ambient dimensions and introduce a CPD solving strategy that leverages fixing a factor and solving a structured linear system, augmented by a greedy scheme to maximize monomial columns. These results establish exact, guaranteed CPDs over finite fields and illuminate the landscape of tensor Decompositions from a fixed-parameter perspective, with implications for exact algorithms in algebraic settings and potential extensions to broader domains like integers. The work also opens questions about optimal constants for fixed parameters and the scope of FPT results beyond finite fields.

Abstract

We present a simple proof that finding a rank- canonical polyadic decomposition of a 3-dimensional tensor over a finite field is fixed-parameter tractable with respect to and . We also show a nontrivial upper bound on the time complexity of this problem.
Paper Structure (15 sections, 10 theorems, 9 equations, 1 table, 2 algorithms)

This paper contains 15 sections, 10 theorems, 9 equations, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Proof of fixed-parameter tractability
  4. Tighter time complexity
  5. Pre- and post-processing
  6. CPD solving
  7. Row reduction
  8. Refinement of matrix factorization
  9. Time complexity
  10. Future directions
  11. Fast low-rank matrix factorization
  12. Row reduction for monomial columns
  13. Matrix factorization
  14. Rank-$r$ factorization summands of a rank-$r$ matrix
  15. Rank-$r$ factorization summands of a rank-$(r-1)$ matrix

Key Result

Theorem 1

For whole number $R$ and finite field $\mathbb{F}$, determining whether a rank-$R$ CPD over $\mathbb{F}$ of a $n_0\times n_1\times n_2$ tensor exists, and constructing such a CPD if it exists, can be done in $O(f(|\mathbb{F}|,R)+\mathrm{poly}(n_0,n_1,n_2,R))$ time for some function $f$ and $O(\mathr

Theorems & Definitions (16)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 6 more