A new mathematical embedding method designed to preserve both the geometry of data and its direction has matched known movement patterns in several synthetic tests, according to an arXiv preprint. In a Swiss-roll experiment, the recovered drift, a vector representing local directional movement, had a cosine alignment of 1.00 with the true drift, meaning the two directions matched in the reported measure. The relative error in estimated asymmetry ranged from 0.07 to 0.17, and the error decreased as the sample size grew.
At its core, the paper asks whether Finsler geometry, a framework that allows direction to be part of a space's geometry, can model directed data while preserving both underlying geometry and directionality. In plain terms, it seeks an embedding that retains where points sit and the way relationships run, rather than reducing the data to an undirected picture.
Keeping geometry and direction together
The construction separates the signal into two parts. As the bandwidth used to compare nearby points becomes small, the symmetric operator approaches a weighted Laplacian, a diffusion-like operator that describes geometric structure. The antisymmetric operator approaches a first-order transport operator that carries the directional information.
Those statements rely on a specific mathematical setting. The analysis assumes independent, identically distributed samples from a smooth, connected and compact Finsler manifold without a boundary. The samples come from a smooth density that is bounded away from zero, and the kernel used by the method must be smooth, positive, decreasing and have a sub-exponential tail.
Under a shrinking bandwidth and the stated relationship between sample size and bandwidth, the discrete symmetric and antisymmetric operators converge uniformly and almost surely to their continuous counterparts. That is an asymptotic guarantee under the paper's conditions, rather than a finite-sample promise for every dataset.
From theory to an embedding
The proposed procedure accepts directed distances or a weighted directed adjacency matrix, together with kernel and embedding settings. It returns embedded samples, drift vectors and Randers metrics. In the recovery argument, the antisymmetric operator recovers the centroid field pushed into embedding space, while the symmetric operator's carré du champ, a matrix derived from the symmetric part, recovers the Binet-Legendre metric in those coordinates. Those two recovered ingredients are then used to construct a moment-Randers approximation.
Synthetic tests
The first quantitative demonstration used a Swiss roll. The study treated a two-dimensional manifold embedded in three-dimensional space as a Randers manifold, sampled 2,000 points, connected them with a 10-neighbor nearest-neighbor graph and used Isomap for the embedding. Across the reported configurations, the drift cosine was 1.00 and the relative asymmetry error stayed between 0.07 and 0.17.
A separate Matsumoto experiment sampled 3,000 points, built a 10-neighbor directed nearest-neighbor graph with GEORCE edge weights and embedded the points in two dimensions with Isomap. The recovered drift aligned with the gradient of h. The paper presents this as a qualitative result without a numerical alignment score, saying the pattern suggests that the method captures the main directional structure of a non-Randers metric.
In a directed stochastic block model, the method was tested on 15 communities and 1,000 nodes. The reported configuration used a within-community probability of 0.1, a forward probability of 0.4 and a backward probability of 0.5, with a two-dimensional embedding. Across flow settings, the recovered drift followed the known direction when the forward and backward probabilities differed: the median signed cosine was plus or minus 0.89 when their absolute difference was 0.02, and above 0.99 from a difference of 0.1 onward. When the probabilities were equal, there was no preferred direction, while the measured asymmetry was symmetric and grew linearly with the size of the probability difference.
Evidence still limited
The evidence is mathematical and asymptotic, supported by demonstrations on synthetic data. The reproducibility statement says the datasets can be regenerated and the code is publicly available under the MIT License. The reported results therefore show how the construction behaves in the stated models and assumptions, but do not establish its performance on real-world directed datasets.
The paper also says its appendix contains the proofs and assumptions behind the analysis. Its experiments support the proposed framework within those settings, while leaving its performance on broader directed data untested.
Paper data and sources
Original title: A Finslerian Approach for Embedding Directed Data
Authors: Gwendal Debaussart-Joniec, Théau Blanchard, Argyris Kalogeratos
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-09-29
DOI: Not available
Original paper · Full text