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

TL;DR

This work introduces Finsler MDS to address asymmetric dissimilarities by embedding data into a simple canonical Randers space, where the asymmetry is controlled by a drift vector ω and geodesics remain straight. It extends the SMACOF optimization to the Finsler setting with theoretical convergence guarantees and demonstrates neural-network-based embedding learning for large digraphs. Empirically, Finsler MDS yields intuitive visualizations and superior node embeddings and link prediction on directed graphs, outperforming traditional Euclidean MDS. The framework broadens manifold learning by integrating asymmetric geometry, enabling richer representations of real-world data with directional relationships.

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.

Figures (8)

• Figure 1: External fields, such as non-uniform currents, lead to time-wise geodesic curves from $A$ to $B$ (in yellow) that differ from those from $B$ to $A$ (in red). Although asymmetry is impossible in Riemannian manifolds, it is at the core of Finsler geometry.
• Figure 2: Flattening the Swiss roll (left) equipped with a Randers metric (middle) in the 3D canonical Randers space with Finsler MDS (right). The plotted arrows on the manifold are the linear drift components $\tilde{\omega}$ of the Randers metric. While Finsler MDS manages to flatten the Swiss roll, it preserves the asymmetry along $\omega$, e.g. the height difference between blue and red points implies asymmetric distances between them.
• Figure 3: Swiss roll embeddings to the canonical Randers space $\mathbb{R}^2$. Our Finsler MDS can generalise current SOTA approaches for providing embeddings robust to missing parts. Note that Finsler MDS accurately embeds the Swiss roll to $\mathbb{R}^2$, but Isomap would not be able to accurately embed the symmetric version to $\mathbb{R}^1$.
• Figure 4: Unflattening a current map (left) by Finsler MDS embedding to the 3D canonical Randers space (right). Dissimilarities are shortest-time distances given the current. The 3D map reveals the asymmetry, where timewise distances are easy to read based on the measurements on the straight Euclidean 3D line between points, and with local maxima (resp. minima), plotted in black (resp. gray), corresponding to source (resp. sink) points.
• Figure 5: Manually rotated embeddings of modified directed binary trees of various depth, with small symmetric edges between equally deep nodes. The reference works, t-SNE van2008visualizing and UMAP mcinnes2018umap fail to provide a meaningful embedding. The Fruchterman-Reingold fruchterman1991graph algorithm fails to clearly reveal the hierarchy, unlike Isomap schwartz1989numericaltenenbaum2000global. The latter heavily distorts distances between equally deep nodes, due to node collapse to two modes and large separations at the top of the structure. Neither method provides the direction of the hierarchy. In contrast, the Finsler MDS embedding clearly reveals the hierarchy and its direction by construction while highly preserving all distances. The symbols mean: M -- Metric, AA -- Asymmetric algorithm, and AE -- Asymmetric Embedding.

Theorems & Definitions (14)

• Definition 1: Finsler metric
• Definition 2: Randers metric
• Definition 3: Canonical Randers space
• Theorem 1: Flatness of the canonical Randers space
• Proposition 1: Canonical Randers distance
• Theorem 2
• Proposition 2: Finsler SMACOF
• Theorem 3: Euler-Lagrange equation
• Lemma 1
• Proposition 3