[論文レビュー] Harnessing Data Asymmetry: Manifold Learning in the Finsler World

Thomas Dagès, Simon Weber+2 more|arXiv (Cornell University)|Mar 12, 2026

Morphological variations and asymmetry被引用数 0

ひとこと要約

The paper proposes a Finsler-based manifold learning pipeline that constructs and embeds asymmetric dissimilarities using Randers metrics, extending t-SNE and UMAP to asymmetric data and demonstrating improved embeddings over Euclidean baselines on synthetic and real datasets.

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 suggest to harness this asymmetry by switching to Finsler geometry, an asymmetric generalisation of Riemannian geometry, and propose a Finsler manifold learning pipeline that constructs asymmetric dissimilarities and embeds in a Finsler space. This greatly broadens the applicability of existing asymmetric embedders beyond traditionally directed data to any data. We also modernise asymmetric embedders by generalising current reference methods to asymmetry, like Finsler t-SNE and Finsler Umap. On controlled synthetic and large real datasets, we show that our asymmetric pipeline reveals valuable information lost in the traditional pipeline, e.g. density hierarchies, and consistently provides superior quality embeddings than their Euclidean counterparts.

研究の動機と目的

• Reveal the inconsistency of traditional symmetric data constructions in manifold learning.
• Embrace sampling-induced asymmetry by using a Finsler metric to enrich data representations.
• Embed data in a canonical Finsler space and generalise modern embedders like t-SNE and UMAP to asymmetric settings.
• Develop scalable, asymmetric embedding methods with efficient optimisation.
• Demonstrate the practical benefits of asymmetry-aware embeddings on synthetic and real datasets.

提案手法

• Construct asymmetric dissimilarities from data via local metric scaling and density-aware transforms without symmetrisation.
• Embed into a canonical Randers (Finsler) space to capture directional asymmetry.
• Generalise modern embedding methods to asymmetric data by replacing Euclidean distances with Finsler distances in the embedding objectives.
• Derive explicit gradients and update rules for Finsler t-SNE and Finsler UMAP to enable scalable optimisation.
• Improve computational efficiency by adapting to sparse dissimilarities and gradient-based optimisation.
• Provide theoretical justification for asymmetry-aware data construction and its impact on embeddings.

実験結果

リサーチクエスチョン

• RQ1 How does sampling-induced asymmetry affect traditional manifold learning pipelines that assume symmetry?
• RQ2 Can Finsler geometry, specifically Randers metrics, effectively encode and utilise asymmetry in data dissimilarities during embedding?
• RQ3 Do asymmetric Finsler embeddings (Finsler t-SNE, Finsler UMAP) outperform symmetric Euclidean embeddings on synthetic and real datasets in terms of clustering and representation quality?
• RQ4 How can modern embedding techniques be adapted to handle asymmetric data at scale?
• RQ5 What additional information, such as density hierarchies, can be recovered via asymmetric embeddings that is lost in symmetric pipelines?

主な発見

• Asymmetric dissimilarities constructed from data reveal density-related structure that is not captured by symmetric embeddings.
• Finsler embeddings consistently outperform Euclidean baselines on label-related clustering metrics across multiple datasets.
• Finsler t-SNE and Finsler UMAP provide scalable optimisations with explicit gradients in the Randers embedding framework.
• Density hierarchies and cluster representations emerge more clearly in Finsler embeddings than in symmetric methods.
• Experiments on synthetic and real datasets, including US cities and image classification benchmarks, show improved embedding quality with the proposed approach.