Harnessing Data Asymmetry: Manifold Learning in the Finsler World
arXiv:2603.11396v1 Announce Type: new Abstract: 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 asymme
arXiv:2603.11396v1 Announce Type: new Abstract: 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.
Executive Summary
This article proposes a novel approach to manifold learning by harnessing data asymmetry using Finsler geometry. The authors argue that traditional methods relying on symmetric Riemannian geometry discard valuable asymmetric information inherent to non-uniform data samples. They introduce a Finsler manifold learning pipeline that constructs asymmetric dissimilarities and embeds in a Finsler space, thereby broadening the applicability of existing asymmetric embedders. The proposed approach is evaluated on controlled synthetic and large real datasets, demonstrating its ability to reveal valuable information lost in traditional pipelines and provide superior quality embeddings. This work has significant implications for data analysis and visualization, particularly in domains where data is inherently asymmetric or non-uniform.
Key Points
- ▸ The article proposes a Finsler manifold learning pipeline to harness data asymmetry
- ▸ The approach constructs asymmetric dissimilarities and embeds in a Finsler space
- ▸ The method is evaluated on controlled synthetic and large real datasets
Merits
Strength in theoretical foundation
The article provides a rigorous theoretical foundation for Finsler manifold learning, drawing on principles from Finsler geometry and Riemannian geometry.
Effective evaluation on diverse datasets
The authors demonstrate the efficacy of their approach on a range of controlled synthetic and large real datasets, showcasing its ability to reveal valuable information and provide high-quality embeddings.
Demerits
Limited experimentation on complex, real-world scenarios
While the authors evaluate their approach on diverse datasets, the article could benefit from more experimentation on complex, real-world scenarios to further establish its practical utility.
Potential computational complexity
The Finsler manifold learning pipeline may introduce computational complexity due to the asymmetric nature of Finsler geometry, which could impact its scalability and practicality.
Expert Commentary
The article makes a significant contribution to the field of manifold learning by introducing a novel approach to harnessing data asymmetry. The authors' use of Finsler geometry to construct asymmetric dissimilarities and embed in a Finsler space is a compelling solution to the limitations of traditional methods. While the article is theoretically sound, further experimentation on complex, real-world scenarios is necessary to establish the practical utility of the proposed approach. Nevertheless, the article's findings have significant implications for a range of domains, from recommender systems to policy-making.
Recommendations
- ✓ Future research should focus on developing scalable and efficient algorithms for Finsler manifold learning to address potential computational complexity issues.
- ✓ The authors should further investigate the application of their approach to complex, real-world scenarios to demonstrate its practical utility and robustness.