A Geometric Theory of Higher-Order Automatic Differentiation
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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.
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