Depth-first search for tensor rank and border rank over finite fields

2024/11/22 by Jason Yang, Yang, Jason
Computer Science · Mathematics · #Algorithms and Data Compression #Coding theory and cryptography #Computational Complexity (cs.CC) #FOS: Computer and information sciences #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2411.14676

openalex publication_date 2024/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present an O^*(|\mathbbF|^{(R-n_*)(∑_d n_d)+n_*})-time algorithm for determining whether a tensor of shape n₀×…× n_D-1 over a finite field \mathbbF has rank ≤ R, where n_*:=max_d n_d; we assume without loss of generality that ∀ d:n_d≤ R. We also extend this problem to its border rank analog, i.e., determining tensor rank over rings of the form \mathbbF[x]/(x^H), and give an O^*(|\mathbbF|^H∑_{1≤ r≤ R} ∑_d min(r,n_d))-time algorithm. Both of our algorithms use polynomial space.

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