@article{culbertson_bayesian_2013,
title = {Bayesian machine learning via category theory},
url = {http://arxiv.org/abs/1312.1445},
abstract = {From the Bayesian perspective, the category of conditional probabilities (a variant of the Kleisli category of the Giry monad, whose objects are measurable spaces and arrows are Markov kernels) gives a nice framework for conceptualization and analysis of many aspects of machine learning. Using categorical methods, we construct models for parametric and nonparametric Bayesian reasoning on function spaces, thus providing a basis for the supervised learning problem. In particular, stochastic processes are arrows to these function spaces which serve as prior probabilities. The resulting inference maps can often be analytically constructed in this symmetric monoidal weakly closed category. We also show how to view general stochastic processes using functor categories and demonstrate the Kalman filter as an archetype for the hidden Markov model.},
urldate = {2019-11-22},
journal = {arXiv:1312.1445 [math]},
author = {Culbertson, Jared and Sturtz, Kirk},
month = dec,
year = {2013},
note = {ZSCC: 0000006
arXiv: 1312.1445},
keywords = {Bayesianism, Categorical ML, Categorical probability theory, Purely theoretical}
}
@book{engeler_combinatory_1995,
series = {Progress in {Theoretical} {Computer} {Science}},
title = {The {Combinatory} {Programme}},
isbn = {978-0-8176-3801-6},
url = {https://www.springer.com/gb/book/9780817638016},
abstract = {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.},
language = {en},
urldate = {2019-11-26},
publisher = {Birkhäuser Basel},
author = {Engeler, Erwin},
year = {1995},
doi = {10.1007/978-1-4612-4268-0},
note = {ZSCC: NoCitationData[s1] },
keywords = {Algebra, Programming language theory, Purely theoretical}
}
@article{fong_backprop_2019,
title = {Backprop as {Functor}: {A} compositional perspective on supervised learning},
shorttitle = {Backprop as {Functor}},
url = {http://arxiv.org/abs/1711.10455},
abstract = {A supervised learning algorithm searches over a set of functions \$A {\textbackslash}to B\$ parametrised by a space \$P\$ to find the best approximation to some ideal function \$f{\textbackslash}colon A {\textbackslash}to B\$. It does this by taking examples \$(a,f(a)) {\textbackslash}in A{\textbackslash}times B\$, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent---with respect to a fixed step size and an error function satisfying a certain property---defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.},
urldate = {2019-11-23},
journal = {arXiv:1711.10455 [cs, math]},
author = {Fong, Brendan and Spivak, David I. and Tuyéras, Rémy},
month = may,
year = {2019},
note = {ZSCC: 0000015
arXiv: 1711.10455},
keywords = {Categorical ML, Machine learning, Purely theoretical}
}
@article{mccullagh_what_2002,
title = {What is a statistical model?},
volume = {30},
url = {http://projecteuclid.org/euclid.aos/1035844977},
doi = {10/bkts3m},
language = {en},
number = {5},
urldate = {2019-11-22},
journal = {The Annals of Statistics},
author = {McCullagh, Peter},
month = oct,
year = {2002},
note = {ZSCC: 0000230},
keywords = {Bayesianism, Categorical ML, Categorical probability theory, Compendium, Purely theoretical, Statistical learning theory},
pages = {1225--1310}
}
@misc{watanabe_algebraic_2009,
title = {Algebraic {Geometry} and {Statistical} {Learning} {Theory}},
url = {/core/books/algebraic-geometry-and-statistical-learning-theory/9C8FD1BDC817E2FC79117C7F41544A3A},
abstract = {Cambridge Core - Pattern Recognition and Machine Learning - Algebraic Geometry and Statistical Learning Theory - by Sumio Watanabe},
language = {en},
urldate = {2019-11-22},
journal = {Cambridge Core},
author = {Watanabe, Sumio},
month = aug,
year = {2009},
doi = {10.1017/CBO9780511800474},
note = {ZSCC: 0000276 },
keywords = {Algebra, Bayesianism, Purely theoretical, Statistical learning theory}
}