Related papers: Cartesian Difference Categories
Integer-order differential operators were originally used to describe local and isotropic effects, in both space and time. However, in fields like biology, the modelling of complex phenomena with spatial heterogeneity necessitates more…
This paper provides a compositional approach to Taylor expansion, in the setting of cartesian differential categories. Taylor expansion is captured here by a functor that generalizes the tangent bundle functor to higher order derivatives.…
Causal inference is often portrayed as fundamentally distinct from predictive modeling, with its own terminology, goals, and intellectual challenges. But at its core, causal inference is simply a structured instance of prediction under…
Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information.…
The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the…
The category of Cartesian cubical sets is introduced and endowed with a Quillen model structure using ideas coming from recent constructions of cubical systems of univalent type theory.
Causal functions of sequences occur throughout computer science, from theory to hardware to machine learning. Mealy machines, synchronous digital circuits, signal flow graphs, and recurrent neural networks all have behaviour that can be…
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…
The present paper gives a generalization of cartesian closed categories, called cartesian closed categories with dependence, whose strict version induces categories with families that support 1-, Sigma- and Pi-types in the strict sense.…
We present a systematic recipe for generating and classifying duality transformations in one-dimensional quantum lattice systems. Our construction emphasizes the role of global symmetries, including those described by (non)-abelian groups…
This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number…
Coherent differentiation was introduced by Ehrhard in order to generalize differential categories to a setting in which the sum is only partially defined, in order to account for the deterministic nature of most models of computation. This…
We give a rigorous formulation of the intuitive idea that a differentiable map should be thesame thing as a locally, or infinitesimally, linear map: just as a linear map respects the operations of addition and multiplication by scalars ina…
Automatic differentiation, as implemented today, does not have a simple mathematical model adapted to the needs of modern machine learning. In this work we articulate the relationships between differentiation of programs as implemented in…
A distinguished variety is a variety that exits the bidisk through the distinguished boundary. We show that Ando's inequality for commuting matrix contractions can be sharpened to looking at the maximum modulus on a distinguished variety,…
The Causal Bandit is a variant of the classic Bandit problem where an agent must identify the best action in a sequential decision-making process, where the reward distribution of the actions displays a non-trivial dependence structure that…
EFRST20, the notion of $c$-differentials was introduced as a potential expansion of differential cryptanalysis against block ciphers utilizing substitution boxes. Drawing inspiration from the technique of higher order differential…
Classification is a well-studied machine learning task which concerns the assignment of instances to a set of outcomes. Classification models support the optimization of managerial decision-making across a variety of operational business…
For a discrete function $f\left( x\right) $ on a discrete set, the finite difference can be either forward and backward. However, we observe that if $ f\left( x\right) $ is a sum of two functions $f\left( x\right) =f_{1}\left( x\right)…
We define a differential Tannakian category and show that under a natural assumption it has a fibre functor. If in addition this category is neutral, that is, the target category for the fibre functor are finite dimensional vector spaces…