@article{martin-maroto_algebraic_2018,
title = {Algebraic {Machine} {Learning}},
url = {http://arxiv.org/abs/1803.05252},
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 parameter-free, 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, hand-written 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 bottom-up and top-down information, formal concept derivation from data and large-scale parallelization.},
urldate = {2019-10-10},
journal = {arXiv:1803.05252 [cs, math]},
author = {Martin-Maroto, Fernando and de Polavieja, Gonzalo G.},
month = mar,
year = {2018},
note = {arXiv: 1803.05252}
}
@article{murfet_logic_2019,
title = {Logic and the \$2\$-{Simplicial} {Transformer}},
url = {http://arxiv.org/abs/1909.00668},
abstract = {We introduce the \$2\$-simplicial Transformer, an extension of the Transformer which includes a form of higher-dimensional attention generalising the dot-product attention, and uses this attention to update entity representations with tensor products of value vectors. We show that this architecture is a useful inductive bias for logical reasoning in the context of deep reinforcement learning.},
urldate = {2019-11-21},
journal = {arXiv:1909.00668 [cs, stat]},
author = {Murfet, Daniel and Clift, James and Doryn, Dmitry and Wallbridge, James},
month = sep,
year = {2019},
note = {ZSCC: 0000000
arXiv: 1909.00668
version: 1},
keywords = {Abstract machines, Algebra, Machine learning, Semantics}
}
@article{philipona_is_2003,
title = {Is {There} {Something} {Out} {There}? {Inferring} {Space} from {Sensorimotor} {Dependencies}},
volume = {15},
shorttitle = {Is {There} {Something} {Out} {There}?},
doi = {10/frg7gs},
abstract = {This letter suggests that in biological organisms, the perceived structure of reality, in particular the notions of body, environment, space, object, and attribute, could be a consequence of an effort on the part of brains to account for the dependency between their inputs and their outputs in terms of a small number of parameters. To validate this idea, a procedure is demonstrated whereby the brain of a (simulated) organism with arbitrary input and output connectivity can deduce the dimensionality of the rigid group of the space underlying its input-output relationship, that is, the dimension of what the organism will call physical space.},
journal = {Neural computation},
author = {Philipona, David and O’Regan, J. and Nadal, Jean-Pierre},
month = oct,
year = {2003},
note = {ZSCC: 0000225},
keywords = {Algebra, Neuroscience},
pages = {2029--49}
}