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.

翻译：机器学习已逐步推广至非欧几里得领域，但在曲面上进行几何精确学习的方法仍显滞后。缺乏闭式黎曼算子、其离散对应物的不可微性以及较差的并行化能力，一直是该领域在网格上发展的主要障碍。最直测地线为计算离散化为网格的黎曼曲面上的指数映射提供了一个原理性框架，同时也能作为副产品追踪测地线和平移向量。我们提供了一个并行的GPU实现，并推导了两种不同的通过最直测地线进行微分的方法：一种利用外部代理函数，另一种基于测地线有限差分方案。在证明了我们的并行化性能和准确性之后，我们展示了我们的可微分指数映射如何改进一般几何形状上的学习和优化流程。特别是，为了展示我们方法的通用性，我们提出了一种新的测地线卷积层、一种用于网格学习的新流匹配方法，以及一个我们应用于中心Voronoi剖分的二阶优化器。我们的代码、模型和可通过pip安装的库（digeo）可在以下网址获取：circle-group.github.io/research/DSG。