A Finslerian Approach for Embedding Directed Data

2026-09-29

AI Summary

This study addresses the limitation of existing spectral embedding methods in preserving directional information when processing directed data. To overcome this, we model directed data as samples from a Finsler manifold and introduce asymmetric Finsler distances with Randers metrics. By leveraging the moment expansion of kernel operators, our approach decouples geometric structure from directional information, accompanied by rigorous proofs of operator convergence. This work integrates Finsler geometry, differential operator theory, and graph signal processing to innovatively extract explicit directional vector fields within a spectral clustering framework. Experiments on synthetic directed graphs and point cloud data demonstrate that the proposed method successfully recovers underlying manifold structures and drift fields, effectively bridging the theoretical gaps inherent in conventional spectral approaches.

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.

Problem

• directed data
• spectral embedding
• asymmetry
• Finsler manifold
• directionality

Innovation

• Finsler manifold
• directed data embedding
• spectral embedding
• kernel operator
• Randers metrics