# Change the Product, Keep the Parameters: Associative Algebra Layers for Transformers

arXiv: https://arxiv.org/abs/2609.32814 · HTML: https://mlsanity.com/paper/2609.32814/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 24 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 15 of 15)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
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- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Instruction-Following Evaluation for Large Language Models](https://mlsanity.com/paper/2311.07911/) (2023)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Monarch: Expressive Structured Matrices for Efficient and Accurate Training](https://mlsanity.com/paper/2204.00595/) (2022)
- [Beyond Fully-Connected Layers with Quaternions: Parameterization of Hypercomplex Multiplications with 1/n Parameters](https://mlsanity.com/paper/2102.08597/) (2021)
- [Compute Better Spent: Replacing Dense Layers with Structured Matrices](https://mlsanity.com/paper/2406.06248/) (2024)
- [Scalable MatMul-free Language Modeling](https://mlsanity.com/paper/2406.02528/) (2024)
- [Algebraic Neural Networks: Stability to Deformations](https://mlsanity.com/paper/2009.01433/) (2020)
- [AlgebraNets](https://mlsanity.com/paper/2006.07360/) (2020)
- [Improving the matrix multiplication exponent with modern optimization and AlphaEvolve](https://mlsanity.com/paper/2608.16884/) (2026)
- [Fully tensorial approach to hypercomplex-valued neural networks](https://mlsanity.com/paper/2407.00449/) (2024)

## Abstract (arXiv)

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.