arXiv：cs.LG· Alex Guerra, Joshua Aduol, Yaqian Tang, Daniel Williams, Ryan P. Adams, Peter Orbanz·· 11 小时前
Coordinate-Free Automatic Differentiation：一种无需固定坐标基的自动微分算子
Coordinate-Free Automatic Differentiation
AI 导读
研究者提出一种与基无关的导数算子，可统一表达旋转、体积形式、曲率等多种几何微分运算，并完全用现有自动微分原语实现。该变换可与 JAX 的批处理、编译和硬件加速组合，将几何作为代数数据引入，并在流形算子复用、时空逆问题流水线和转子子群优化上得到实验验证。
正文
Abstract: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.
| Comments: | 34 pages, 18 figures |
| Subjects: | Machine Learning (cs.LG); Mathematical Software (cs.MS); Differential Geometry (math.DG) |
| Cite as: | arXiv:2610.12776 [cs.LG] |
| (or arXiv:2610.12776v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2610.12776 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Alex Guerra [view email]
[v1]
Thu, 8 Oct 2026 22:35:10 UTC (7,019 KB)
来源：arXiv：cs.LG · arxiv.org