Irreversibility of Structure Tensors of Modules

TL;DR

This paper proves that certain algebraic approaches cannot resolve the matrix multiplication exponent , highlighting fundamental limitations in current methods from algebra and complexity theory.

Contribution

It establishes the irreversibility of using structure tensors of modules to prove , extending previous results and combining techniques from algebra and complexity theory.

Findings

Proves impossibility of proving =2 using structure tensors of modules

Shows limitations of starting with -generic non-diagonal tensors

Generalizes previous work by Ble4ser and Lysikov

Abstract

Determining the matrix multiplication exponent $\omega$ is one of the greatest open problems in theoretical computer science. We show that it is impossible to prove $\omega = 2$ by starting with structure tensors of modules of fixed degree and using arbitrary restrictions. It implies that the same is impossible by starting with $1_A$-generic non-diagonal tensors of fixed size with minimal border rank. This generalizes the work of Bl\"aser and Lysikov [3]. Our methods come from both commutative algebra and complexity theory.