# Algebraic Integer Quantization for Machine Learning Applications

arXiv: https://arxiv.org/abs/2610.12494 · HTML: https://mlsanity.com/paper/2610.12494/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 20 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 7 of 7)

- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [The LAMBADA dataset: Word prediction requiring a broad discourse context](https://mlsanity.com/paper/1606.06031/) (2016)
- [QReg: On Regularization Effects of Quantization](https://mlsanity.com/paper/2206.12372/) (2022)
- [Emulation of Complex Matrix Multiplication based on the Chinese Remainder Theorem](https://mlsanity.com/paper/2512.08321/) (2025)
- [Precision Neural Network Quantization via Learnable Adaptive Modules](https://mlsanity.com/paper/2504.17263/) (2025)

## Abstract (arXiv)

In this article, we propose algebraic integer number formats (AINF) as a framework for highly configurable and accurate compressed computation. A key property of AINF is its straightforward reduction to purely integer arithmetic, enabling error-free multiply-accumulate operations. We develop a matrix multiplication engine that exploits this property, obtaining performance within a predictable factor of the most efficient numerical counterparts on modern hardware accelerators. Another key advantage of AIQ is the freedom to select algebraic integers tailored to the numerical characteristics of a given use case, offering optimization opportunities that are fundamentally unavailable in floating-point or fixed-point formats. Motivated by these properties, we introduce algebraic integer quantization (AIQ), a lossy compression format for large language model (LLM) inference, and provide results demonstrating that AIQ achieves competitive accuracy compared to the industry-standard MX format across a diverse set of models and benchmarks at equivalent bit-per-parameter budgets. The experiments presented in this article employ algebraic integers of degree two (quadratic integers); generalizations to higher-order and, in particular, multidimensional algebraic integers promise even greater computational advantages and remain a promising direction for future work.