用代数分解重构学习，让模型直接从数据中推导出规则。
Algebraic Machine Learning: Learning as computing an algebraic decomposition of a task
- 将任务和数据编码为代数公理，通过子直分解提取原子结构。
- 在多个数据集上性能接近优化过的多层感知机。
- 适合追求可解释性与理论严谨性的研究者。
统计与优化是现代机器学习的基础。本文提出一种基于抽象代数的新基础，使学习过程更易于分析。该方法将任务目标与数据编码为代数公理，得到的模型仅满足这些公理及其逻辑推论。尽管此模型不具备泛化能力，但我们证明：通过选择其代数原子分解中的特定子集，可获得具有泛化能力的模型。我们在标准数据集如MNIST、FashionMNIST、CIFAR-10及医学图像上验证了该学习原则，性能可与优化后的多层感知机相媲美。此外，该原则还可扩展至形式问题，例如仅凭规范即可求解哈密顿回路，无需搜索。该代数基础为机器智能提供了新视角：直接从训练数据学习，无需验证集，支持模型可加性扩展，并能渐近收敛至数据中的底层规律。
原文摘要 · Abstract (English)
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.
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