# Harnessing Data Asymmetry: Manifold Learning in the Finsler World

arXiv: https://arxiv.org/abs/2603.11396 · HTML: https://mlsanity.com/paper/2603.11396/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 133 references that are in the map.
- First posted on arXiv 12 Mar 2026.

## Builds on (most cited first, 23 of 23)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Describing Textures in the Wild](https://mlsanity.com/paper/1311.3618/) (2013)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [Hyperbolic Disk Embeddings for Directed Acyclic Graphs](https://mlsanity.com/paper/1902.04335/) (2019)
- [Hyperbolic Diffusion Embedding and Distance for Hierarchical Representation Learning](https://mlsanity.com/paper/2305.18962/) (2023)
- [Spectral Clustering Based on Local PCA](https://mlsanity.com/paper/1301.2007/) (2013)
- [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)
- [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)
- [Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding](https://mlsanity.com/paper/2503.18010/) (2025)
- [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)
- [IAN: Iterated Adaptive Neighborhoods for manifold learning and dimensionality estimation](https://mlsanity.com/paper/2208.09123/) (2022)
- [Curvature Augmented Manifold Embedding and Learning](https://mlsanity.com/paper/2403.14813/) (2024)
- [Inductive Global and Local Manifold Approximation and Projection](https://mlsanity.com/paper/2406.08097/) (2024)
- [IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations](https://mlsanity.com/paper/2407.17835/) (2024)

## Built on by (most cited first, 1 of 1)

- [A Finslerian Approach for Embedding Directed Data](https://mlsanity.com/paper/2609.37649/) (2026)

## Abstract (arXiv)

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.