Algebraic Machine Learning: learning as computing subsets of the subdirect decomposition from Abstract Algebra

We propose a machine-learning framework based on the principle that learning can be achieved through algebraic decomposition rather than numerical optimization. A learning task is encoded as axioms of an algebra, and learning proceeds by computing a subdirect decomposition of the algebra into subdirectly irreducible components. We show that suitable subsets of these components form models that generalize to unseen data because they approximate the underlying rule in the training data. The method has no dataset-specific hyperparameters to tune and shows no evidence of classical overfitting in our experiments, so we run it without a validation dataset. We compare against multilayer perceptrons as generic parametric baselines without task-specific inductive biases. Across 24 classification datasets, the algebraic method is applied only once to each training set without hyperparameter tuning, whereas the multi-layer perceptrons are selected from many configurations using validation data or cross-validation. Nevertheless, the resulting test accuracies are statistically indistinguishable. Moreover, adding a statistical readout on top of the algebraic representation, using untuned logistic regression, improves over the best multilayer perceptrons selected by validation or cross-validation. More broadly, because task axioms may encode examples, formal knowledge, or both, this approach provides a common algebraic framework and the same algorithm for data-driven problems such as classification, formally specified problems such as computing Hamiltonian cycles from specifications, and mixed settings combining data with formal constraints.