TY - BOOK
TI - The Combinatory Programme
AU - Engeler, Erwin
T2 - Progress in Theoretical Computer Science
AB - Combinatory logic started as a programme in the foundation of mathematics and in an historical context at a time when such endeavours attracted the most gifted among the mathematicians. This small volume arose under quite differ ent circumstances, namely within the context of reworking the mathematical foundations of computer science. I have been very lucky in finding gifted students who agreed to work with me and chose, for their Ph. D. theses, subjects that arose from my own attempts 1 to create a coherent mathematical view of these foundations. The result of this collaborative work is presented here in the hope that it does justice to the individual contributor and that the reader has a chance of judging the work as a whole. E. Engeler ETH Zurich, April 1994 lCollected in Chapter III, An Algebraization of Algorithmics, in Algorithmic Properties of Structures, Selected Papers of Erwin Engeler, World Scientific PubJ. Co. , Singapore, 1993, pp. 183-257. I Historical and Philosophical Background Erwin Engeler In the fall of 1928 a young American turned up at the Mathematical Institute of Gottingen, a mecca of mathematicians at the time; he was a young man with a dream and his name was H. B. Curry. He felt that he had the tools in hand with which to solve the problem of foundations of mathematics mice and for all. His was an approach that came to be called "formalist" and embodied that later became known as Combinatory Logic.
DA - 1995///
PY - 1995
DP - www.springer.com
LA - en
PB - Birkhäuser Basel
SN - 978-0-8176-3801-6
UR - https://www.springer.com/gb/book/9780817638016
Y2 - 2019/11/26/14:23:14
KW - Algebra
KW - Programming language theory
KW - Purely theoretical
ER -
TY - CONF
TI - Algebraic classifiers: a generic approach to fast cross-validation, online training, and parallel training
AU - Izbicki, Michael
AB - We use abstract algebra to derive new algorithms for fast cross-validation, online learning, and parallel learning. To use these algorithms on a classification model, we must show that the model has appropriate algebraic structure. It is easy to give algebraic structure to some models, and we do this explicitly for Bayesian classifiers and a novel variation of decision stumps called HomStumps. But not all classifiers have an obvious structure, so we introduce the Free HomTrainer. This can be used to give a "generic" algebraic structure to any classifier. We use the Free HomTrainer to give algebraic structure to bagging and boosting. In so doing, we derive novel online and parallel algorithms, and present the first fast cross-validation schemes for these classifiers.
C3 - ICML
DA - 2013///
PY - 2013
DP - Semantic Scholar
ST - Algebraic classifiers
KW - Algebra
KW - Categorical ML
KW - Machine learning
ER -
TY - JOUR
TI - Algebraic Machine Learning
AU - Martin-Maroto, Fernando
AU - de Polavieja, Gonzalo G.
T2 - arXiv:1803.05252 [cs, math]
AB - 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.
DA - 2018/03/14/
PY - 2018
DP - arXiv.org
UR - http://arxiv.org/abs/1803.05252
Y2 - 2019/10/10/11:42:39
ER -
TY - SLIDE
TI - Algebra and Artiﬁcial Intelligence
A2 - Murfet, Daniel
LA - en
KW - Algebra
KW - Classical ML
KW - Machine learning
KW - Sketchy
ER -
TY - COMP
TI - dmurfet/2simplicialtransformer
AU - Murfet, Daniel
AB - Code for the 2-simplicial Transformer paper. Contribute to dmurfet/2simplicialtransformer development by creating an account on GitHub.
DA - 2019/10/14/T08:10:47Z
PY - 2019
DP - GitHub
LA - Python
UR - https://github.com/dmurfet/2simplicialtransformer
Y2 - 2019/11/22/16:50:05
KW - Abstract machines
KW - Algebra
KW - Implementation
KW - Machine learning
KW - Semantics
ER -
TY - JOUR
TI - Logic and the $2$-Simplicial Transformer
AU - Murfet, Daniel
AU - Clift, James
AU - Doryn, Dmitry
AU - Wallbridge, James
T2 - arXiv:1909.00668 [cs, stat]
AB - 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.
DA - 2019/09/02/
PY - 2019
DP - arXiv.org
UR - http://arxiv.org/abs/1909.00668
Y2 - 2019/11/21/20:31:14
KW - Abstract machines
KW - Algebra
KW - Machine learning
KW - Semantics
ER -
TY - JOUR
TI - Is There Something Out There? Inferring Space from Sensorimotor Dependencies
AU - Philipona, David
AU - O’Regan, J.
AU - Nadal, Jean-Pierre
T2 - Neural computation
AB - 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.
DA - 2003/10/01/
PY - 2003
DO - 10/frg7gs
DP - ResearchGate
VL - 15
SP - 2029
EP - 49
J2 - Neural computation
ST - Is There Something Out There?
KW - Algebra
KW - Neuroscience
ER -
TY - ELEC
TI - Algebraic Geometry and Statistical Learning Theory
AU - Watanabe, Sumio
T2 - Cambridge Core
AB - Cambridge Core - Pattern Recognition and Machine Learning - Algebraic Geometry and Statistical Learning Theory - by Sumio Watanabe
DA - 2009/08//
PY - 2009
LA - en
UR - /core/books/algebraic-geometry-and-statistical-learning-theory/9C8FD1BDC817E2FC79117C7F41544A3A
Y2 - 2019/11/22/18:05:57
KW - Algebra
KW - Bayesianism
KW - Purely theoretical
KW - Statistical learning theory
ER -