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

2025-02-27

AI Summary

This paper proposes a novel paradigm for machine learning grounded in abstract algebra, aiming to transcend the limitations of statistical modeling and numerical optimization by unifying data-driven tasks and formal reasoning problems—such as deriving Hamiltonian cycles from logical specifications. Methodologically, task objectives and data are encoded as axioms within an algebraic system; generalizable models are then constructed directly via subdirect decomposition, eliminating the need for validation-set-based hyperparameter tuning, enabling additive model composition, and guaranteeing asymptotic convergence to the intrinsic structure of the data. The core contribution is the first systematic adoption of subdirect decomposition as the foundational mechanism of learning, enabling purely symbolic, axiom-to-model derivation. Experiments demonstrate competitive performance against tuned multilayer perceptrons on MNIST, Fashion-MNIST, CIFAR-10, and medical image classification; moreover, the framework successfully solves specification-driven combinatorial problems without explicit search.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Proposes algebraic foundation for machine learning
• Encodes tasks and data as algebraic axioms
• Extends learning to formal problems without search

Innovation

• Abstract Algebra foundation
• Subdirect decomposition model
• Direct learning without validation