Reverse Tangent Categories

TL;DR

This paper introduces reverse tangent categories, extending reverse differential categories to more general spaces like smooth manifolds, enabling gradient-based learning in broader mathematical contexts.

Contribution

The paper proposes reverse tangent categories, a new framework that generalizes reverse differential categories to include smooth manifolds with an involution operation.

Findings

Defines reverse tangent categories with involution on differential bundles.

Extends gradient-based learning to functions on smooth manifolds.

Provides a categorical framework for advanced geometric learning methods.

Abstract

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.