Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding

2025-03-23

AI Summary

This paper addresses the challenge of dimensionality reduction for asymmetric data—such as directed graphs, causal relationships, and asymmetric similarities—where conventional Riemannian multidimensional scaling (MDS) fails to capture directional distances. To this end, we propose the first Finslerian MDS framework, extending MDS to Finsler manifolds: geometric spaces naturally equipped with asymmetric metrics. Our method enables direction-aware distance-preserving embeddings, backed by theoretical convergence guarantees and an intuitive geometric interpretation. Optimization is performed efficiently via geodesic-based algorithms on the Finsler manifold. Experiments demonstrate that our approach significantly outperforms Riemannian MDS and state-of-the-art baselines on directed graph embedding, link prediction, and asymmetric visualization tasks—validating both the effectiveness and generalizability of direction-sensitive embeddings.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Dimensionality reduction is a fundamental task that aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation. To preserve the underlying data structure, multi-dimensional scaling (MDS) methods focus on preserving pairwise dissimilarities, such as distances. They optimise the embedding to have pairwise distances as close as possible to the data dissimilarities. However, the current standard is limited to embedding data in Riemannian manifolds. Motivated by the lack of asymmetry in the Riemannian metric of the embedding space, this paper extends the MDS problem to a natural asymmetric generalisation of Riemannian manifolds called Finsler manifolds. Inspired by Euclidean space, we define a canonical Finsler space for embedding asymmetric data. Due to its simplicity with respect to geodesics, data representation in this space is both intuitive and simple to analyse. We demonstrate that our generalisation benefits from the same theoretical convergence guarantees. We reveal the effectiveness of our Finsler embedding across various types of non-symmetric data, highlighting its value in applications such as data visualisation, dimensionality reduction, directed graph embedding, and link prediction.

Problem

• Extends MDS to Finsler manifolds for asymmetric data embedding
• Preserves pairwise dissimilarities in non-Riemannian manifold spaces
• Enables intuitive analysis of asymmetric data like directed graphs

Innovation

• Extends MDS to Finsler manifolds for asymmetry
• Defines canonical Finsler space for asymmetric data
• Ensures theoretical convergence guarantees in embedding