Finsler geometry improves embedding of directional data sets
A Finslerian Approach for Embedding Directed Data
Machine Learning
Summary
Many data sets have a sense of direction, like traffic flowing on roads or citations pointing backward in time. Traditional methods often ignore this direction by treating data symmetrically, losing important information. The authors propose using a Finsler manifold, a mathematical space where distance depends on direction, to better capture this. Their approach separates the shape of the data from its direction and provides a way to embed directed data while preserving its asymmetry. They tested their method on simulated data to demonstrate its effectiveness.
What this means in practice
- •For network analysts: Embed and analyze directed networks like citation graphs or traffic flows while preserving direction information in their structure.
- •For bioinformatics teams: Map cell differentiation data along lineages capturing both the shape and directionality of biological processes.
Tested on simulated data.
Authors
Gwendal Debaussart-Joniec, Théau Blanchard, Argyris Kalogeratos
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.