A Finslerian Approach for Embedding Directed Data
Abstract
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.
Related
BibTeX
@misc{debaussartjoniec2026finslerian,
title = {{A Finslerian Approach for Embedding Directed Data}},
author = {Gwendal Debaussart-Joniec and Théau Blanchard and Argyris Kalogeratos},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37649},
eprint = {2609.37649},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05769186v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A Finslerian Approach for Embedding Directed Data
AU - Gwendal Debaussart-Joniec
AU - Théau Blanchard
AU - Argyris Kalogeratos
PY - 2026
DA - 2026/09/29
PB - HAL
DO - 10.48550/arxiv.2609.37649
UR - https://hal.science/hal-05769186v1
AB - 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.
ER -