A Geometric Theory of Higher-Order Automatic Differentiation

Understand

First-order automatic differentiation is a ubiquitous tool across statistics, machine learning, and computer science.

• Higher-order implementations of automatic differentiation, however, have yet to realize the same utility.
• In this paper I derive a comprehensive, differential geometric treatment of automatic differentiation that naturally identifies the higher-order differential operators amenable to automatic differentiation as well as explicit procedures that provide a scaffolding for high-performance implementations.