2025/02/27 by Fernando Martín-Maroto, Martin-Maroto, Fernando, Nabil Abderrahaman +5
Computer Science · #03G10 #06A06 #06A12 #08A70 #68R01 #68T01 #Advanced Graph Neural Networks #Artificial Intelligence (cs.AI) #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #G.2.3 #I.1.2 #I.2.6 #I.2.8 #Machine Learning (cs.LG) #Polynomial and algebraic computation #Symbolic Computation (cs.SC) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2502.19944
openalex publication_date 2025/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.