# Algebraic Machine Learning: Learning as computing an algebraic decomposition of a task

arXiv: https://arxiv.org/abs/2502.19944 · HTML: https://mlsanity.com/paper/2502.19944/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2025.

## Built on by (most cited first, 1 of 1)

- [Algebraic Machine Learning for Small-to-Medium Datasets Is Competitive against Strong Standard Baselines](https://mlsanity.com/paper/2605.22155/) (2026)

## Abstract (arXiv)

Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the task and the data are encoded as axioms of an algebra, and a model is obtained where only these axioms and their logical consequences hold. Although this is not a generalizing model, we show that selecting specific subsets of its breakdown into algebraic atoms obtained via subdirect decomposition gives a model that generalizes. We validate this new learning principle on standard datasets such as MNIST, FashionMNIST, CIFAR-10, and medical images, achieving performance comparable to optimized multilayer perceptrons. Beyond data-driven tasks, the new learning principle extends to formal problems, such as finding Hamiltonian cycles from their specifications and without relying on search. This algebraic foundation offers a fresh perspective on machine intelligence, featuring direct learning from training data without the need for validation dataset, scaling through model additivity, and asymptotic convergence to the underlying rule in the data.