This paper introduces Finsler geometry to manifold learning, allowing for the capture of asymmetric data relationships like density hierarchies that Riemannian methods ignore.
March 13, 2026
Original Paper
Harnessing Data Asymmetry: Manifold Learning in the Finsler World
arXiv · 2603.11396
The Takeaway
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