This paper introduces Finsler geometry to manifold learning, allowing for the capture of asymmetric data relationships like density hierarchies that Riemannian methods ignore.
arXiv · March 13, 2026 · 2603.11396
Why it matters
Standard manifold learning (t-SNE, UMAP) assumes symmetric distances (Riemannian/Euclidean), which discards information in non-uniform datasets. This framework generalizes embedding techniques to asymmetric spaces, significantly improving the visualization and analysis of complex, directed data structures.
From the abstract
Manifold learning is a fundamental task at the core of data analysis and visualisation. It aims to capture the simple underlying structure of complex high-dimensional data by preserving pairwise dissimilarities in low-dimensional embeddings. Traditional methods rely on symmetric Riemannian geometry, thus forcing symmetric dissimilarities and embedding spaces, e.g. Euclidean. However, this discards in practice valuable asymmetric information inherent to the non-uniformity of data samples. We sugg