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

arXiv: https://arxiv.org/abs/2503.18010 · HTML: https://mlsanity.com/paper/2503.18010/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 125 references that are in the map.
- First posted on arXiv 23 Mar 2025.

## Builds on (most cited first, 25 of 25)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [word2vec Explained: deriving Mikolov et al.'s negative-sampling word-embedding method](https://mlsanity.com/paper/1402.3722/) (2014)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Edge Directionality Improves Learning on Heterophilic Graphs](https://mlsanity.com/paper/2305.10498/) (2023)
- [Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach](https://mlsanity.com/paper/2106.04941/) (2021)
- [Magnetic eigenmaps for community detection in directed networks](https://mlsanity.com/paper/1606.07359/) (2016)
- [Directed Graph Auto-Encoders](https://mlsanity.com/paper/2202.12449/) (2022)
- [HoloNets: Spectral Convolutions do extend to Directed Graphs](https://mlsanity.com/paper/2310.02232/) (2023)
- [Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning](https://mlsanity.com/paper/2305.18962/) (2023)
- [Node Representation Learning for Directed Graphs](https://mlsanity.com/paper/1810.09176/) (2018)
- [Wormhole Loss for Partial Shape Matching](https://mlsanity.com/paper/2410.22899/) (2024)
- [Intrinsic Isometric Manifold Learning with Application to Localization](https://mlsanity.com/paper/1806.00556/) (2018)
- [On Unsupervised Partial Shape Correspondence](https://mlsanity.com/paper/2310.14692/) (2023)
- [Generalized Nonlinear and Finsler Geometry for Robotics](https://mlsanity.com/paper/2010.14745/) (2020)
- [Finsler-Laplace-Beltrami Operators with Application to Shape Analysis](https://mlsanity.com/paper/2404.03999/) (2024)
- [DUPLEX: Dual GAT for Complex Embedding of Directed Graphs](https://mlsanity.com/paper/2406.05391/) (2024)
- [Tree-Wasserstein Distance for High Dimensional Data with a Latent Feature Hierarchy](https://mlsanity.com/paper/2410.21107/) (2024)
- [Low-dimensional statistical manifold embedding of directed graphs](https://mlsanity.com/paper/1905.10227/) (2019)
- [Metric Convolutions: A Unifying Theory to Adaptive Image Convolutions](https://mlsanity.com/paper/2406.05400/) (2024)

## Built on by (most cited first, 8 of 8)

- [Learning Lineage-guided Geodesics with Finsler Geometry](https://mlsanity.com/paper/2603.16708/) (2026)
- [Harnessing Data Asymmetry: Manifold Learning in the Finsler World](https://mlsanity.com/paper/2603.11396/) (2026)
- [Finsler Geometry, Graph Neural Networks, and You](https://mlsanity.com/paper/2606.17185/) (2026)
- [Learning Eigenstructures of Unstructured Data Manifolds](https://mlsanity.com/paper/2512.01103/) (2025)
- [GLACIER: A Multimodal Student-Teacher Foundation Model for Molecular Property Prediction](https://mlsanity.com/paper/2606.11382/) (2026)
- [mathbbSL(n) Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition](https://mlsanity.com/paper/2609.15083/) (2026)
- [RGSQ: Riemannian Geometry-Sensitive Quantization for Large Vision-Language Models](https://mlsanity.com/paper/2609.25492/) (2026)
- [EDISCO: Equivariant DIScrete Diffusion for Euclidean Combinatorial Optimization](https://mlsanity.com/paper/2610.04953/) (2026)

## Abstract (arXiv)

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.