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Backprop as Functor: A compositional perspective on supervised learning

Brendan Fong, David Spivak, Remy Tuyeras
2019 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
Backpropagation gives a way to compute the derivative of a function via message passing on a network, significantly speeding up learning.  ...  This request function allows us to construct training data for all sub-parts of a composite learning algorithm from training data for just the input and output of the composite algorithm.  ... 
doi:10.1109/lics.2019.8785665 dblp:conf/lics/FongST19 fatcat:g7btwbi5frfjliyghc4bkiaakq

Backprop as Functor: A compositional perspective on supervised learning [article]

Brendan Fong, David I. Spivak, Rémy Tuyéras
2019 arXiv   pre-print
A supervised learning algorithm searches over a set of functions A → B parametrised by a space P to find the best approximation to some ideal function f A → B.  ...  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  ...  quicker to compute for lower dimensional spaces, expresses the speed up in learning provided by backpropagation.  ... 
arXiv:1711.10455v3 fatcat:m2xi2jtrgrfo7adviem23zopxq

CLASSIFYING PROCESSES: AN ESSAY IN APPLIED ONTOLOGY

Barry Smith
2012 Ratio  
the first steps of an approach to the classification of such processes within the BFO framework. 1  ...  BFO is a bicategorial ontology, embracing both three-dimensionalist (continuant) and four-dimensionalist (occurrent) perspectives within a single framework.  ...  First Approximation to a Solution of the Problem of Process Measurement Data How, then, do we respond to the need on the part of the users of BFO to annotate data deriving from measurements which have  ... 
doi:10.1111/j.1467-9329.2012.00557.x pmid:23888086 pmcid:PMC3718480 fatcat:ecgsa3rf6jbfpdk6rbrijz6lmq

Classifying Processes: An Essay in Applied Ontology [chapter]

Barry Smith
2013 Classifying Reality  
First Approximation to a Solution of the Problem of Process Measurement Data How, then, do we respond to the need on the part of the users of BFO to annotate data deriving from measurements which have  ...  when we talk, for example, of the speed of light, or the speed of the earth, or the speed of a billiard ball.  ... 
doi:10.1002/9781118627747.ch6 fatcat:f72xdelfvzab5e4hvccyozzvhy

Lenses and Learners [article]

Brendan Fong, Michael Johnson
2019 arXiv   pre-print
Lenses are a well-established structure for modelling bidirectional transformations, such as the interactions between a database and a view of it.  ...  More recently, the notion of a learner has been proposed: these provide a compositional way of modelling supervised learning algorithms, and again form the morphisms of a monoidal category.  ...  BF thanks Jules Hedges for first bringing the contents of Remark 4.7 to his attention.  ... 
arXiv:1903.03671v2 fatcat:ot2vkmpognabpgrqci2helzg4e

Programs: Improvements, Complexity, and Meanings (Dagstuhl Seminar 98231)

Neil D. Jones, Oege de Moor, James S. Royer
1998
The organizers wish to thank all the participants, as well has the Dagstuhl staff, for making this a successful meeting.  ...  A considerable amount of bridge building took place, but there were clearly some chasms no one yet knows how to cross (e.g., how to deal with all the issues involved in rigorous complexity analyses of  ...  We present a new application of Berry and Curien's intensional semantics of sequential algorithms on concrete data structures and its related programming language, CDS0.  ... 
doi:10.4230/dagsemrep.213 fatcat:j3wgtbf5ufb6flypjmweaswdsu

Seven Sketches in Compositionality: An Invitation to Applied Category Theory [article]

Brendan Fong, David I Spivak
2018 arXiv   pre-print
The tour takes place over seven sketches, each pairing an evocative application, such as databases, electric circuits, or dynamical systems, with the exploration of a categorical structure, such as adjoint  ...  This book is an invitation to discover advanced topics in category theory through concrete, real-world examples. It aims to give a tour: a gentle, quick introduction to guide later exploration.  ...  We want to draw the graph corresponding to the instance I : G → Set shown below: Email sent_by received_by Em_1 Bob Grace Em_2 Grace Pat Em_3 Bob Emory Em_4 Sue Doug Em_5 Doug Sue Em_6 Bob Bob  ... 
arXiv:1803.05316v3 fatcat:mpg54meyrngepe3ohm7i2h5pt4

Short Steps in Noncommutative Geometry [article]

Ahmad Zainy Al-Yasry
2019 arXiv   pre-print
A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure in which one  ...  Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative  ...  up to a 2-isomorphism.  ... 
arXiv:1901.03640v1 fatcat:usxvglooabfg5hntp2ep2j2qtu

This Week's Finds in Mathematical Physics (1-50) [article]

John C. Baez
2022 arXiv   pre-print
These issues focus on quantum gravity, topological quantum field theory, knot theory, and applications of n-categories to these subjects.  ...  These are the first 50 issues of This Week's Finds of Mathematical Physics, from January 19, 1993 to March 12, 1995.  ...  We introduce a path-dependent hamiltonian representation (the path representation) for SU (2) with fermions in 3 + 1 dimensions.  ... 
arXiv:2101.04168v3 fatcat:kh5eurusrjevbkttpuq24p3s4q

This Week's Finds in Mathematical Physics (51-100) [article]

John C. Baez
2022 arXiv   pre-print
These are the second 50 issues of This Week's Finds of Mathematical Physics, from April 23, 1995 to March 23, 1997.  ...  But now suppose one wanted to build a computer language that could not only deal with all sorts of data types corresponding to different "sets with essentially algebraic structure", but also various "categories  ...  Mathematically sophisticated computer scientists want to be able define data types corresponding to arbitrary sorts of sets with essentially algebraic structure, and to play around with them easily.  ... 
arXiv:2203.02631v1 fatcat:mhrjnnmee5a7nljvh4jt4qyou4

Behavioral Congruences and Verification of Graph Transformation Systems with Applications to Model Refactoring [article]

Guilherme Salum Rangel, Technische Universität Berlin, Technische Universität Berlin, Hartmut Ehrig
2008
This algorithm forms the core of a tool support which will enable us to come up with behavior analysi [...]  ...  The concept of borrowed contexts has opened up graph transformations to the notion of an external "observer" where graphs (specifying systems) may interact with an environment in order to evolve.  ...  Proof Techniques for DPO-BC with NACs We introduce here some proof techniques to speed up the bisimulation checking procedure.  ... 
doi:10.14279/depositonce-2032 fatcat:xjkmz7g575gyfegfsja3gmn6bq