Research map / Learning theory and algorithms / Prediction and decision losses
Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Extends multi-dimensional scaling from Riemannian to Finsler (asymmetric) embedding spaces by introducing a canonical Randers space, a Finsler SMACOF solver, and a neural Finsler digraph embedding evaluated on visualisation and link prediction.
Problem
Traditional MDS embeds asymmetric dissimilarities into Riemannian (symmetric-distance) spaces, so it cannot capture asymmetric pairwise dissimilarities such as those in directed graphs or flows with external fields.
Approach
Generalise the MDS embedding space from Riemannian to Finsler geometry, using a canonical Randers space R^m with Euclidean metric plus a constant drift vector; geodesics stay straight segments while distances become asymmetric via a closed-form term, and the stress is minimised with a Finsler SMACOF iteration or a Fermi-Dirac neural decoder.
Key results
- On six digraphs, Finsler embeddings give lower distortion and higher MAP than Euclidean embeddings at dimensions 2, 5, 10 and 50 (e.g., Cora MAP 0.894 vs 0.574 at m=2).
- For link existence prediction the Finsler method surpasses competing methods on most datasets (e.g., Cora 95.2 vs DUPLEX 95.0 AUC), and is competitive on Squirrel.
- For link direction prediction it achieves the best AUC on most datasets (e.g., Arxiv-Year 74.4 vs 70.2 for MagNet).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Finsler representation learning instead of DUPLEX | Cora link existence prediction · Cora | ROC AUC: 95.0±0.2 -> 95.2±0.3 | Helped |
| Use Finsler representation learning instead of DUPLEX | Citeseer link existence prediction · Citeseer | ROC AUC: 97.2±0.6 -> 94.6±1.0 | Hurt |
| Use Finsler representation learning instead of DUPLEX | Gr-QC link existence prediction · Gr-QC | ROC AUC: 83.1±0.3 -> 84.9±0.6 | Helped |
| Use Finsler representation learning instead of DUPLEX | Chameleon link existence prediction · Chameleon | ROC AUC: 87.2±0.4 -> 89.3±0.8 | Helped |
| Use Finsler representation learning instead of MagNet | Squirrel link existence prediction · Squirrel | ROC AUC: 82.6±1.0 -> 82.3±0.3 | Hurt |
| Use Finsler representation learning instead of MagNet | Arxiv-Year link existence prediction · Arxiv-Year | ROC AUC: 65.8±1.3 -> 68.6±1.4 | Helped |
| Use Finsler representation learning instead of DUPLEX | Cora link direction prediction · Cora | ROC AUC: 95.3±0.3 -> 95.1±0.2 | Hurt |
| Use Finsler representation learning instead of DUPLEX | Citeseer link direction prediction · Citeseer | ROC AUC: 97.9±0.1 -> 97.2±0.3 | Hurt |
| Use Finsler representation learning instead of DUPLEX | Gr-QC link direction prediction · Gr-QC | ROC AUC: 90.7±0.1 -> 92.6±0.5 | Helped |
| Use Finsler representation learning instead of DUPLEX | Chameleon link direction prediction · Chameleon | ROC AUC: 92.0±0.4 -> 93.2±0.4 | Helped |
| Use Finsler representation learning instead of MagNet | Squirrel link direction prediction · Squirrel | ROC AUC: 85.2±0.8 -> 89.0±0.3 | Helped |
| Use Finsler representation learning instead of MagNet | Arxiv-Year link direction prediction · Arxiv-Year | ROC AUC: 70.2±2.4 -> 74.4±1.8 | Helped |
Limitations
- No closed-form solution for Finsler MDS; relies on iterative descent reaching only local minima.
- Distortion grows as data departs from the chosen metric structure.
- Canonical Randers space encodes all asymmetry in a single dimension, which may be insufficient for some applications.
Experimental setup
Datasets: Cora Citeseer Gr-QC Chameleon Squirrel Arxiv-Year Swiss roll (synthetic) modified directed binary trees (synthetic)
Benchmarks: Cora Citeseer Gr-QC Chameleon Squirrel Arxiv-Year
Metrics: normalised stress (distortion) MAP ROC AUC
Baselines: Isomap t-SNE UMAP Fruchterman-Reingold NERD DiGCN MagNet DiGAE ODIN DUPLEX
Builds on: SMACOF Isomap Fermi-Dirac decoder
Closest papers by meaning
- Harnessing Data Asymmetry in Manifold Learning (NeurIPS 2026 · similarity 0.95)
- Euclidean Embedding of Data Using Local Distances (NeurIPS 2026 · similarity 0.88)
- Partition First, Embed Later: Laplacian-Based Feature Partitioning for Refined Embedding and Visualization of High-Dimensional Data (ICML 2025 · similarity 0.87)
- Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models (NeurIPS 2025 · similarity 0.87)
- Neuc-MDS: Non-Euclidean Multidimensional Scaling Through Bilinear Forms (NeurIPS 2024 · similarity 0.87)
- Toward a Unified Geometry Understanding : Riemannian Diffusion Framework for Graph Generation and Prediction (NeurIPS 2025 · similarity 0.86)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.