Coordinate-Free Automatic Differentiation
We show that (i) a single, basis-independent derivative operator can represent many geometric differential operations, and (ii) this operator can be expressed entirely in terms of existing AD primitives.
ProofPaper ↗
Key points
- Automatic differentiation (AD) is a cornerstone of modern machine learning.
- Since all AD frameworks implicitly fix a basis of Euclidean space to define gradients and Jacobians, they natively process mathematical objects that are arrays of a fixed set of coordinates.
- This makes it awkward to express objects such as rotations, volume forms, and curvatures that arise, for example, in applications of AI to science.
- We experimentally demonstrate its reuse across different manifold operators, its composition through a spacetime inverse-problem pipeline, and its use in optimization over different rotor subgroups.
Sources (1)
- [1]Coordinate-Free Automatic DifferentiationarXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 8, 10:35 PM
We show that (i) a single, basis-independent derivative operator can represent many geometric differential operations, and (ii) this operator can be expressed entirely in terms of existing AD primitives.
Automatic differentiation (AD) is a cornerstone of modern machine learning.
Extractive summary: sentences quoted from the sources.