Reverse tangent categories

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 learning for functions between more general spaces such as smooth manifolds. In this paper we propose a setting to handle this, which we call reverse tangent categories: tangent categories with an involution operation for their differential bundles

Reverse Derivative Categories

The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.Comment: Extended version of paper to appear at CSL 202

Introduction to tangent categories

In this talk I'll introduce the idea of a tangent category, which can be seen as a minimal categorical setting for differential geometry. I'll discuss a variety of examples, and then focus on how analogs of vector spaces and (affine) connections can be defined in any tangent category. Time-permitting, I'll also briefly describe a few other structures that can be defined in a tangent category, including differential forms and (ordinary) differential equations and their solutions.Non UBCUnreviewedAuthor affiliation: Mount Allison UniversityResearche

Category Theory for Cognitive Science

Applied Category Theory (aCT) is both a language for describing and a method for analyzing the abstract structures that populate cognitive science. It can serve as the lingua franca for cross domain discussions, and as a mathematical tool for probing the consequences of a model or theory’s structure. As many cognitive scientists are unfamiliar with aCT this workshop will provide an introduction to its terminology and key features. The morning session will emphasize key concepts and tutorial exercises. The afternoon session will use recent research applications as case studies of its potential and as the basis for demonstrations of how it can be used productively