Topological Data Analysis (TDA) utilise topology-inspired invariants, most notably persistent homology, to extract structural features from complex datasets. A fundamental requirement for these invariants in computational applications is stability under spatial perturbations. Within the standard setting of TDA, the Gromov–Hausdorff distance serves as a rigorous metric framework for comparing underlying datasets and establishing stability guarantees, ensuring that small metric deformations result in bounded changes in the corresponding persistence diagrams.

While classical TDA focuses primarily on the geometric arrangement of unlabelled point clouds, modern applications frequently require integrating qualitative features, usually represented by different colours, directly onto the points of a dataset. A prominent example is bioimages of tissues, where different cell types are represented in different colours. This necessity has driven the emergence of chromatic TDA. To capture the homological interactions between distinct coloured subsets, recent techniques utilise the “six-pack”, a collection of six interlinked persistence diagrams.

In this talk, we recall the standard framework of TDA and the role of the Gromov–Hausdorff distance. We then present some of the techniques utilised to study and compute features from these coloured datasets. Finally, we introduce the $C$-constrained Gromov–Hausdorff distance, a suitable variation of the classical metric adapted for chromatic frameworks, and demonstrate its application in evaluating invariants in chromatic TDA—specifically by establishing the stability of the six-pack.

O. Draganov, S. Rosenmeier, N. Zava, Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack, arXiv:2507.17994.