# TopoMap++: A faster and more space efficient technique to compute projections with topological guarantees

arXiv: https://arxiv.org/abs/2409.07257 · HTML: https://mlsanity.com/paper/2409.07257/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 58 references that are in the map.
- First posted on arXiv 11 Sep 2024.

## Builds on (most cited first, 12 of 12)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Revisiting Dimensionality Reduction Techniques for Visual Cluster Analysis: An Empirical Study](https://mlsanity.com/paper/2110.02894/) (2021)
- [TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data](https://mlsanity.com/paper/2009.01512/) (2020)
- [Improving Metric Dimensionality Reduction with Distributed Topology](https://mlsanity.com/paper/2106.07613/) (2021)
- [Homology-Preserving Dimensionality Reduction via Manifold Landmarking and Tearing](https://mlsanity.com/paper/1806.08460/) (2018)
- [Attribute-based Explanations of Non-Linear Embeddings of High-Dimensional Data](https://mlsanity.com/paper/2108.08706/) (2021)
- [Topologically Regularized Data Embeddings](https://mlsanity.com/paper/2110.09193/) (2021)
- [Topology-Preserving Dimensionality Reduction via Interleaving Optimization](https://mlsanity.com/paper/2201.13012/) (2022)
- [Untangling Force-Directed Layouts Using Persistent Homology](https://mlsanity.com/paper/2208.06927/) (2022)

## Built on by (most cited first, 3 of 3)

- [Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction](https://mlsanity.com/paper/2502.20215/) (2025)
- [When One Point Is Not Enough: Addressing Ambiguous Instances in Dimensionality Reduction by Splitting](https://mlsanity.com/paper/2605.23540/) (2026)
- [FADEx: Feature Attribution and Distortion-based Explanation of Dimensionality Reduction](https://mlsanity.com/paper/2607.27463/) (2026)

## Abstract (arXiv)

High-dimensional data, characterized by many features, can be difficult to visualize effectively. Dimensionality reduction techniques, such as PCA, UMAP, and t-SNE, address this challenge by projecting the data into a lower-dimensional space while preserving important relationships. TopoMap is another technique that excels at preserving the underlying structure of the data, leading to interpretable visualizations. In particular, TopoMap maps the high-dimensional data into a visual space, guaranteeing that the 0-dimensional persistence diagram of the Rips filtration of the visual space matches the one from the high-dimensional data. However, the original TopoMap algorithm can be slow and its layout can be too sparse for large and complex datasets. In this paper, we propose three improvements to TopoMap: 1) a more space-efficient layout, 2) a significantly faster implementation, and 3) a novel TreeMap-based representation that makes use of the topological hierarchy to aid the exploration of the projections. These advancements make TopoMap, now referred to as TopoMap++, a more powerful tool for visualizing high-dimensional data which we demonstrate through different use case scenarios.