Parallelised Differentiable Straightest Geodesics for 3D Meshes

arXiv cs.CV / 3/18/2026

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Key Points

The paper proposes a principled framework to compute the exponential map on discretized Riemannian surfaces (meshes) using parallel GPU implementation, addressing the main barriers of non-differentiability and poor parallelisation.
It introduces two differentiable pathways through the straightest geodesics—an extrinsic proxy function and a geodesic finite differences scheme—enabling backpropagation through geodesic computations.
The authors demonstrate downstream ML benefits on general geometries, including a new geodesic convolutional layer, a flow matching method for learning on meshes, and a second-order optimiser for centroidal Voronoi tessellation.
They release code, models, and a pip-installable library (digeo), with documentation at circle-group.github.io/research/DSG.
Benchmark results are reported showing parallelization performance and accuracy improvements for learning and optimisation tasks on mesh geometries.

Abstract

Machine learning has been progressively generalised to operate within non-Euclidean domains, but geometrically accurate methods for learning on surfaces are still falling behind. The lack of closed-form Riemannian operators, the non-differentiability of their discrete counterparts, and poor parallelisation capabilities have been the main obstacles to the development of the field on meshes. A principled framework to compute the exponential map on Riemannian surfaces discretised as meshes is straightest geodesics, which also allows to trace geodesics and parallel-transport vectors as a by-product. We provide a parallel GPU implementation and derive two different methods for differentiating through the straightest geodesics, one leveraging an extrinsic proxy function and one based upon a geodesic finite differences scheme. After proving our parallelisation performance and accuracy, we demonstrate how our differentiable exponential map can improve learning and optimisation pipelines on general geometries. In particular, to showcase the versatility of our method, we propose a new geodesic convolutional layer, a new flow matching method for learning on meshes, and a second-order optimiser that we apply to centroidal Voronoi tessellation. Our code, models, and pip-installable library (digeo) are available at: circle-group.github.io/research/DSG.