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Hierarchical topological clustering

Ana Carpio, Gema Duro

TL;DR

The paper tackles clustering of point clouds with arbitrary geometry and meaningful outliers without heavy parameter tuning. It introduces hierarchical topological clustering (HTC) based on persistent homology, specifically $H_0$, using Vietoris-Rips filtrations to generate a hierarchy of clusters as the scale parameter $r$ increases. Through experiments on image, economy, and gene datasets, HTC reveals persistent outliers and interpretable interfaces, often outperforming standard methods like K-means, hierarchical clustering, and DBSCAN. The approach provides a geometrically meaningful, parameter-light framework applicable to diverse domains, including image quality control, trade analysis, and cancer gene research.

Abstract

Topological methods have the potential of exploring data clouds without making assumptions on their the structure. Here we propose a hierarchical topological clustering algorithm that can be implemented with any distance choice. The persistence of outliers and clusters of arbitrary shape is inferred from the resulting hierarchy. We demonstrate the potential of the algorithm on selected datasets in which outliers play relevant roles, consisting of images, medical and economic data. These methods can provide meaningful clusters in situations in which other techniques fail to do so.

Hierarchical topological clustering

TL;DR

The paper tackles clustering of point clouds with arbitrary geometry and meaningful outliers without heavy parameter tuning. It introduces hierarchical topological clustering (HTC) based on persistent homology, specifically , using Vietoris-Rips filtrations to generate a hierarchy of clusters as the scale parameter increases. Through experiments on image, economy, and gene datasets, HTC reveals persistent outliers and interpretable interfaces, often outperforming standard methods like K-means, hierarchical clustering, and DBSCAN. The approach provides a geometrically meaningful, parameter-light framework applicable to diverse domains, including image quality control, trade analysis, and cancer gene research.

Abstract

Topological methods have the potential of exploring data clouds without making assumptions on their the structure. Here we propose a hierarchical topological clustering algorithm that can be implemented with any distance choice. The persistence of outliers and clusters of arbitrary shape is inferred from the resulting hierarchy. We demonstrate the potential of the algorithm on selected datasets in which outliers play relevant roles, consisting of images, medical and economic data. These methods can provide meaningful clusters in situations in which other techniques fail to do so.
Paper Structure (13 sections, 2 equations, 13 figures)

This paper contains 13 sections, 2 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Materials and methods
  3. Hierarchical topological clustering.
  4. HTC Algorithm.
  5. Results and discussion
  6. Clustering fragmented fronts.
  7. Quality assessment in image processing
  8. Outliers in economy datasets
  9. Analysis of gene data
  10. Conclusions
  11. Clustering algorithms
  12. Distances
  13. Image Compression

Figures (13)

  • Figure 1: Scheme of the HTC algorithm.
  • Figure 2: Hierarchical topological clustering applied to a fragmented front separating malignant and healthy cells. Clusters obtained for (a) $r=5.2$, (b) $r=8.8$, (c) $r=9.6$, (d) $r=11$, (e) $r=14.7$, (f) $r=16.9$. The last clusters to merge with the main cluster can be considered collective outliers. They represent islands of malignant cells that have invaded the healthy cells, while the main cluster represents the interface between the two populations.
  • Figure 3: Evolution of the topological clusters with the filtration parameter $r$. We can identify which type of clusters persist for long ranges of $r$.
  • Figure 4: Hierarchical clustering, K-means and DBSCAN applied to the fragmented front. Color representation of (a) the six clusters generated by hierarchical clustering with average linkage and (b) one of the six cluster arrangements obtained by K means (different cluster arrangements are possible). Panel (c) displays the outcome of DBSCAN for $M_p=10$ and $\varepsilon=5$, red points are not assigned to any cluster since they do not have enough neighbors close enough. Compare to the cluster distribution represented in Figure 1(a). The geometrical interpretation of the clusters as representing the main interface and detached islands provided by HTC is lost.
  • Figure 5: Persistence barcode for the $H_0$ homology of the fragmented front dataset. Full barcode (a) versus amplified view of the less persistent clusters (b). This representation illustrates the dynamics of clusters as the filtration parameter varies in a blind way: we do not know which elements belong to each cluster.
  • ...and 8 more figures