Local uniqueness barriers found for complex 48-term matrix multiplication method
Certified local rank and uniqueness barriers for a 48-term matrix-multiplication decomposition
Symbolic ComputationComputational Complexity
Summary
Matrix multiplication can be done in different ways using simpler building blocks, and researchers are interested in the smallest number of these blocks needed. This study looks closely at a known 48-part method for multiplying 4x4 matrices and shows that to find a shorter method, you would have to change many of these parts at once. The authors provide exact measures of how flexible the method is around this known decomposition, meaning any change to use fewer parts faces big barriers. Their work helps understand how unique and stable this matrix multiplication method is on a technical level.
What this means in practice
- •For algorithm designers: Identify limitations on simplifying existing matrix multiplication algorithms by understanding how many basic parts must change when seeking shorter decompositions.
- •For computational complexity engineers: Use local uniqueness and rank radius barriers to evaluate potential improvements on matrix multiplication algorithms in symbolic computation systems.
A theory result. No direct application yet.
Authors
Abhinav Agarwal
Abstract
We study replacements in fixed bilinear tensor decompositions, counting changes to complete rank-one summands, including output factors. The shortening frontier records the maximum rank defect of a fixed-size subset and determines the minimum length attainable within a change budget. For the rational 48-term Li--Wang--Hu decomposition \(D(2)\) of \(4\times4\) matrix multiplication over \(\mathbb{C}\), we prove rank radius at least 12, strong radius exactly 11, and border radius at least 8. Every shorter complex decomposition therefore changes at least thirteen original summands. An exact rational twelve-term replacement attains the equal-length barrier. The proofs combine exhaustive support reductions with saturated projected kernels and zero-corner completion arguments controlling arbitrary minimal competitors. A reduced-incidence argument transfers kernel certificates to tensor-space neighborhoods. A Laurent normal form gives strong radius exactly 11 for the sixteen-term core at every nonzero complex parameter. On a nonempty Zariski-open subset of the actual parameter curve, the rank radius is at least 12, the strong radius exactly 11, and the border radius at least 8. We also prove incomparability of the full Kothari--Moitra--Wein sufficient criterion and the Sylvester-equipped kernel criterion. These results describe local decomposition structure rather than a new rank bound for full matrix multiplication.