# Merging Hazy Sets with m-Schemes: A Geometric Approach to Data Visualization

arXiv: https://arxiv.org/abs/2503.01664 · HTML: https://mlsanity.com/paper/2503.01664/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 21 references that are in the map.
- First posted on arXiv 3 Mar 2025.

## Builds on (most cited first, 3 of 3)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [On UMAP's true loss function](https://mlsanity.com/paper/2103.14608/) (2021)
- [IsUMap: Manifold Learning and Data Visualization leveraging Vietoris-Rips filtrations](https://mlsanity.com/paper/2407.17835/) (2024)

## Abstract (arXiv)

Many machine learning algorithms try to visualize high dimensional metric data in 2D in such a way that the essential geometric and topological features of the data are highlighted. In this paper, we introduce a framework for aggregating dissimilarity functions that arise from locally adjusting a metric through density-aware normalization, as employed in the IsUMap method. We formalize these approaches as m-schemes, a class of methods closely related to t-norms and t-conorms in probabilistic metrics, as well as to composition laws in information theory. These m-schemes provide a flexible and theoretically grounded approach to refining distance-based embeddings.