mlsanityData to 14 Oct 2026

Coordinate-Free Automatic Differentiation

arXiv 2610.12776 ↗First posted 8 Oct 2026cs.LG cs.MS math.DGcited by 0

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Abstract (arXiv)

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. Bespoke workarounds lack reusability. 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. This reusable transformation can be composed with JAX batching, compilation, and hardware acceleration. Geometry enters as algebraic data. 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.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 21 of its 53 references that are in the map. First posted on arXiv 8 Oct 2026.

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References · 21

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
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  11. 2019Trivializations for Gradient-Based Optimization on Manifoldscited by 79
  12. 2023Clifford Group Equivariant Neural Networkscited by 55
9 more
  1. 2023Geometric Algebra Transformercited by 48
  2. 2023Geometric Clifford Algebra Networkscited by 38
  3. 2024Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physicscited by 38
  4. 2024Clifford-Steerable Convolutional Neural Networkscited by 25
  5. 2018The simple essence of automatic differentiationcited by 23
  6. 2024Optimistix: modular optimisation in JAX and Equinoxcited by 19
  7. 2023Euclidean, Projective, Conformal: Choosing a Geometric Algebra for Equivariant Transformerscited by 13
  8. 2024Designing Mechanical Meta-Materials by Learning Equivariant Flowscited by 4
  9. 2025Jet Functors and Weil Algebras in Automatic Differentiation: A Geometric Analysiscited by 1

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