# Enhancing cluster analysis via topological manifold learning

arXiv: https://arxiv.org/abs/2207.00510 · HTML: https://mlsanity.com/paper/2207.00510/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 111 references that are in the map.
- First posted on arXiv 1 Jul 2022.

## Builds on (most cited first, 20 of 20)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Hyperparameter Optimization: Foundations, Algorithms, Best Practices and Open Challenges](https://mlsanity.com/paper/2107.05847/) (2021)
- [A unifying mutual information view of metric learning: cross-entropy vs. pairwise losses](https://mlsanity.com/paper/2003.08983/) (2020)
- [Turning Big data into tiny data: Constant-size coresets for k-means, PCA and projective clustering](https://mlsanity.com/paper/1807.04518/) (2018)
- [ClusterGAN : Latent Space Clustering in Generative Adversarial Networks](https://mlsanity.com/paper/1809.03627/) (2018)
- [Clustering with t-SNE, provably](https://mlsanity.com/paper/1706.02582/) (2017)
- [Topological Data Analysis](https://mlsanity.com/paper/1609.08227/) (2016)
- [Spectral Clustering Based on Local PCA](https://mlsanity.com/paper/1301.2007/) (2013)
- [On the Local Structure of Stable Clustering Instances](https://mlsanity.com/paper/1701.08423/) (2017)
- [TopoMap: A 0-dimensional Homology Preserving Projection of High-Dimensional Data](https://mlsanity.com/paper/2009.01512/) (2020)
- [Validation of cluster analysis results on validation data: A systematic framework](https://mlsanity.com/paper/2103.01281/) (2021)
- [Clustering with UMAP: Why and How Connectivity Matters](https://mlsanity.com/paper/2108.05525/) (2021)
- [A Novel Intrinsic Measure of Data Separability](https://mlsanity.com/paper/2109.05180/) (2021)
- [The SpectACl of Nonconvex Clustering: A Spectral Approach to Density-Based Clustering](https://mlsanity.com/paper/1907.00680/) (2019)
- [Unsupervised Functional Data Analysis via Nonlinear Dimension Reduction](https://mlsanity.com/paper/2012.11987/) (2020)
- [A New Method for Performance Analysis in Nonlinear Dimensionality Reduction](https://mlsanity.com/paper/1711.06252/) (2017)
- [A geometric perspective on functional outlier detection](https://mlsanity.com/paper/2109.06849/) (2021)

## Built on by (most cited first, 6 of 6)

- [Position: Why We Must Rethink Empirical Research in Machine Learning](https://mlsanity.com/paper/2405.02200/) (2024)
- [Beyond Scalars: Concept-Based Alignment Analysis in Vision Transformers](https://mlsanity.com/paper/2412.06639/) (2024)
- [Unsupervised Machine-Learning Pipeline for Data-Driven Defect Detection and Characterisation: Application to Displacement Cascades](https://mlsanity.com/paper/2510.24523/) (2025)
- [Unveiling 3D Ocean Biogeochemical Provinces in the North Atlantic: A Systematic Comparison and Validation of Clustering Methods](https://mlsanity.com/paper/2504.18181/) (2025)
- [CSTS: A Benchmark for the Discovery of Correlation Structures in Time Series Clustering](https://mlsanity.com/paper/2505.14596/) (2025)
- [Assessing the impact of dimensionality reduction on clustering performance - a systematic study](https://mlsanity.com/paper/2604.22099/) (2026)

## Abstract (arXiv)

We discuss topological aspects of cluster analysis and show that inferring the topological structure of a dataset before clustering it can considerably enhance cluster detection: theoretical arguments and empirical evidence show that clustering embedding vectors, representing the structure of a data manifold instead of the observed feature vectors themselves, is highly beneficial. To demonstrate, we combine manifold learning method UMAP for inferring the topological structure with density-based clustering method DBSCAN. Synthetic and real data results show that this both simplifies and improves clustering in a diverse set of low- and high-dimensional problems including clusters of varying density and/or entangled shapes. Our approach simplifies clustering because topological pre-processing consistently reduces parameter sensitivity of DBSCAN. Clustering the resulting embeddings with DBSCAN can then even outperform complex methods such as SPECTACL and ClusterGAN. Finally, our investigation suggests that the crucial issue in clustering does not appear to be the nominal dimension of the data or how many irrelevant features it contains, but rather how textitseparable the clusters are in the ambient observation space they are embedded in, which is usually the (high-dimensional) Euclidean space defined by the features of the data. Our approach is successful because we perform the cluster analysis after projecting the data into a more suitable space that is optimized for separability, in some sense.