Algebraic Machine Learning
Abstract
Machine learning algorithms use error function minimization to fit a large set of parameters in a preexisting model. However, error minimization eventually leads to a memorization of the training dataset, losing the ability to generalize to other datasets. To achieve generalization something else is needed, for example a regularization method or stopping the training when error in a validation dataset is minimal. Here we propose a different approach to learning and generalization that is parameterfree, fully discrete and that does not use function minimization. We use the training data to find an algebraic representation with minimal size and maximal freedom, explicitly expressed as a product of irreducible components. This algebraic representation is shown to directly generalize, giving high accuracy in test data, more so the smaller the representation. We prove that the number of generalizing representations can be very large and the algebra only needs to find one. We also derive and test a relationship between compression and error rate. We give results for a simple problem solved step by step, handwritten character recognition, and the Queens Completion problem as an example of unsupervised learning. As an alternative to statistical learning, algebraic learning may offer advantages in combining bottomup and topdown information, formal concept derivation from data and largescale parallelization.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.05252
 Bibcode:
 2018arXiv180305252M
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Discrete Mathematics;
 Mathematics  Commutative Algebra;
 Mathematics  Rings and Algebras
 EPrint:
 In v2 Figures 10 and 12 are images (v1 used latex commands), so all queens on board are now visible