发表机构
ENS Paris-Saclay; CNRS; Centre Borelli; UMR 1346 HeKA; Inria; Inserm; GE Healthcare（巴黎萨克雷高等师范学院; 法国国家科学研究中心; 博雷利中心; UMR 1346 HeKA实验室; 法国国家信息与自动化研究所; 法国国家健康与医学研究院; 通用电气医疗集团）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于芬斯勒几何的嵌入方法，将有向数据建模为非对称距离流形，通过核算子的对称与反对称部分分离几何与方向，并证明收敛性，在合成数据上验证了其恢复流形结构与漂移的能力。
AI 中文摘要
许多数据集带有内在的方向性：引用指向时间上的过去，细胞沿谱系分化，交通遵循偏好路线。谱嵌入方法，包括其大多数针对有向图的扩展，都丢弃了这一信息：它们对数据进行对称化处理，并将其映射到无法表示非对称性的欧几里得空间。我们转而将有向数据建模为从芬斯勒流形中采样得到，该流形的距离依赖于行进方向，并研究由这种非对称距离构建的核算子。通过对该算子的矩展开，我们证明其对称部分和反对称部分将几何与方向分离开来。当核的带宽趋近于零时，对称部分收敛到加权拉普拉斯算子，在黎曼情形下恢复扩散映射，而反对称部分收敛到编码方向性的一阶输运算子。我们证明由有限样本构建的相应图算子一致且几乎必然地收敛到这些极限。对于兰德斯度量，该向量场是显式的，并产生一种嵌入算法，既能从对称部分的谱中恢复流形结构，又能恢复潜在的漂移。我们在合成有向图和点云上展示了该方法。
英文摘要
Many datasets carry an intrinsic directionality: citations point backward in time, cells differentiate along lineages, and traffic follows preferred routes. Spectral embedding methods, including most of their extensions to directed graphs, discard this information: they symmetrize the data and map it into a Euclidean space where asymmetry cannot be represented. We instead model directed data as sampled from a Finsler manifold, whose distance depends on the direction of travel, and study the kernel operator built from this asymmetric distance. Through a moment expansion of this operator, we show that its symmetric and antisymmetric parts separate geometry from direction. As the bandwidth of the kernel vanishes, the symmetric part converges to a weighted Laplacian, recovering diffusion maps in the Riemannian case, while the antisymmetric part converges to a first-order transport operator that encodes the directionality. We prove that the corresponding graph operators, built from finitely many samples, converge uniformly and almost surely to these limits. For Randers metrics, this vector field is explicit and yields an embedding algorithm recovering both the manifold structure, from the spectrum of the symmetric part, and the underlying drift. We illustrate the approach on synthetic directed graphs and point-clouds.
