2022/09/22 by Filip Rupniewski, Rupniewski, Filip
Computer Science · Mathematics · #14N07 #15A03 (Secondary) #15A69 (Primary) #Algebraic Geometry (math.AG) #Coding theory and cryptography #FOS: Mathematics #Mathematical Approximation and Integration #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2209.11040
openalex publication_date 2022/09/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The article is concerned with the problem of the additivity of the tensor rank. That is for two independent tensors we study when the rank of their direct sum is equal to the sum of their individual ranks. The statement saying that additivity always holds was previously known as Strassen's conjecture (1969) until Shitov proposed counterexamples (2019). They are not explicit and only known to exist asymptotically for very large tensor spaces. In this article, we show that for some small three-way tensors the additivity holds. For instance, we give a proof that another conjecture stated by Strassen (1969) is true. It is the particular case of the general Strassen's additivity conjecture where tensors are a pair of 2 × 2 matrix multiplication tensors. In addition, we show that the Alexeev-Forbes-Tsimerman substitution method preserves the structure of a direct sum of tensors.