Faster search for tensor decomposition over finite fields

Abstract

We present an 𝑂 ∗ (|F| min{𝑅, Í 𝑑≥2 𝑛𝑑 }+(𝑅−𝑛0 ) (Í 𝑑≠0 𝑛𝑑 ) )-time algorithm for determining whether the rank of a concise tensor 𝑇 ∈ F 𝑛0×···×𝑛𝐷−1 is ≤ 𝑅, assuming 𝑛0 ≥ · · · ≥ 𝑛𝐷−1 and 𝑅 ≥ 𝑛0. For 3-dimensional tensors, we have a second algorithm running in 𝑂 ∗ (|F| 𝑛0+𝑛2+(𝑅−𝑛0+1−𝑟∗ ) (𝑛1+𝑛2 )+𝑟 2 ∗ ) time, where 𝑟∗ := j 𝑅 𝑛0 k + 1. Both algorithms use polynomial space and improve on our previous work, which achieved running time 𝑂 ∗ (|F| 𝑛0+(𝑅−𝑛0 ) (Í 𝑑 𝑛𝑑 ) ).