Related papers: Cartesian Differential Storage Categories
Tangent categories provide a categorical axiomatization of the tangent bundle. There are many interesting examples and applications of tangent categories in a variety of areas such as differential geometry, algebraic geometry, algebra, and…
This paper introduces the Token Space framework, a novel mathematical construct designed to enhance the interpretability and effectiveness of deep learning models through the application of category theory. By establishing a categorical…
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here we define a related structure called a monoidal reverse…
The first steps towards linearisation of partial orders and equivalence relations are described. The definitions of partial orders and equivalence relations (on sets) are formulated in a way that is standard in category theory and that…
NVIDIA's CUTLASS library provides a robust and expressive set of methods for describing and manipulating multi-dimensional tensor data on the GPU. These methods are conceptually grounded in the abstract notion of a CuTe layout and a rich…
In this paper we provide an abstract model theory for the untyped differential lambda-calculus and the resource calculus. In particular we propose a general definition of model of these calculi, namely the notion of linear reflexive object…
We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets).…
Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is…
Previous work has shown that reverse differential categories give an abstract setting for gradient-based learning of functions between Euclidean spaces. However, reverse differential categories are not suited to handle gradient-based…
The framework of causal models provides a principled approach to causal reasoning, applied today across many scientific domains. Here we present this framework in the language of string diagrams, interpreted formally using category theory.…
Causal deep learning (CDL) is a new and important research area in the larger field of machine learning. With CDL, researchers aim to structure and encode causal knowledge in the extremely flexible representation space of deep learning…
In regular dynamics, discrete maps are model presentations of discrete dynamical systems, and they may approximate continuous dynamical systems. Maps are used to investigate general properties of dynamical systems and to model various…
Since categories are graphs with additional "structure", one should start from fuzzy graphs in order to define a theory of fuzzy categories. Thus is makes sense to introduce categories whose morphisms are associated with a plausibility…
A new categorical setting is defined in order to characterize the subrecursive classes belonging to complexity hierarchies. This is achieved by means of coercion functors over a symmetric monoidal category endowed with certain recursion…
Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there…
Tensors, or multidimensional arrays, are data structures that can naturally represent visual data of multiple dimensions. Inherently able to efficiently capture structured, latent semantic spaces and high-order interactions, tensors have a…
Cai et al. have recently proposed change structures as a semantic framework for incremental computation. We generalise change structures to arbitrary cartesian categories and propose the notion of change action model as a categorical model…
The term ``Boolean category'' should be used for describing an object that is to categories what a Boolean algebra is to posets. More specifically, a Boolean category should provide the abstract algebraic structure underlying the proofs in…
Neural networks have become an increasingly popular tool for solving many real-world problems. They are a general framework for differentiable optimization which includes many other machine learning approaches as special cases. In this…