| title | Change the Product, Keep the Parameters: Associative Algebra Layers for Transformers |
|---|---|
| authors | Ilya Koziev, Ivan Oseledets |
| date | 2026-09-26 |
| venue | arXiv (preprint, not peer reviewed) |
| id | arXiv:2609.32814 |
| upvotes | 22 (Hugging Face Daily Papers, 2026-11-01) |
| links | abs pdf hf |
| abstract | Fast matrix multiplication algorithms keep the product fixed and search for a cheaper way to evaluate it. We instead ask whether a Transformer's learned projections can use a different, cheaper product altogether. Building on an associative-algebra construction that replaces ordinary matrix multiplication with a sparser interaction table over the same weight blocks, we construct a family with quadratic arithmetic in the matrix dimension when the physical block size remains fixed, and derive finite-shape constraints for GPU execution. The construction is provably optimal for its bilinear rank by the Alder--Strassen bound and can be realized as row-typed rectangular projections compatible with causal masking and KV-cached decoding. We provide an empirical test of this approach by training two approximately 110M-parameter decoder-only Transformer LMs from the same recipe and 12.3B-token budget, differing only in their feed-forward layer: one uses ordinary dense matrix multiplication and the other uses the associative-algebra product. Across four prompt domains, the algebraic model achieves a 6.2--7.8\% increase in end-to-end generation throughput, while obtaining lower scores on all three reported downstream metrics. We treat these results as a feasibility and trainability check for the proposed approach at small scale, leaving further investigation to future work. |
cite
@misc{koziev2026change,
title={Change the Product, Keep the Parameters: Associative Algebra Layers for Transformers},
author={Ilya Koziev and Ivan Oseledets},
year={2026},
eprint={2609.32814},
archivePrefix={arXiv},
url={https://arxiv.org/abs/2609.32814}
}