Chromatic Topological Data Analysis

TL;DR

This survey tackles how to quantify interactions across multiple colored point configurations using topological data analysis. It introduces chromatic $\oldsymbol{\alpha}$-complexes and the chromatic Delaunay mosaic to encode multi-color spatial relations, and packages these insights into the 6-pack of persistent diagrams that jointly describe domain, codomain, kernels, images, and cokernels. A key contribution is the mingling framework, particularly the MST-ratio, which links color mixing to measurable geometric quantities via persistent homology, with rigorous bounds and asymptotics across finite, random, and lattice point sets. The work also develops the algebraic machinery and subcomplex choices needed to extract stable, interpretable descriptors from chromatic data, offering concrete avenues for applications in biology and discrete geometry and pointing to future directions like higher-order colorings, higher dimensions, and stochastic analyses. Overall, the paper provides a comprehensive blueprint for stable, multi-color topological descriptors that quantify complex inter-set interactions at multiple spatial scales, with practical implications for analyzing tumor microenvironments and related discrete-geometric questions.

Abstract

Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial interactions among multiple configurations, each distinguished by a color. It describes advances in this area and prepares the ground for further exploration by mentioning unresolved questions and promising research avenues while focusing on the overlap with discrete geometry.

Figures (5)

• Figure 1: Left: the shaded Delaunay mosaic superimposed on the Voronoi tessellation of a finite set of points. The thick blue edges of the mosaic form the (Euclidean) minimum spanning tree of the points. Middle: the alpha complex is dual to the decomposition of the union of disks into convex regions by the Voronoi tessellation. Right: the corresponding 1-dimensional persistence diagram tracs the birth and death of the loops. Note in particular the two dots far above the diagonal, which represent the two large circles suggested by the points in the set.
• Figure 2: The chromatic Delaunay mosaic of three finite sets in ${\mathbb R}{\hbox{${\mathbb R}$}}^1$ together with a corresponding stratification of space. The points of each set are placed on a copy of ${\mathbb R}{\hbox{${\mathbb R}$}}^1$ orthogonal to the $2$-plane that carries the standard triangle. The stratification consists of Voronoi cells that separate the three colors by forming a $1$-dimensional stratum geometrically located between the three lines, and three $2$-dimensional strata, one between any two of the lines.
• Figure 3: Two empty stacks in ${\mathbb R}{\hbox{${\mathbb R}$}}^2$ that pass through one blue point, two green points, and one orange point forming a simplex in ${\rm Del}{({\chi})}{\hbox{${\rm Del}{({\chi})}$}}$. (In fact, the stack on the right passes through two orange points, so it also passes through the one orange point that lies on the left orange circle.) The set of centers of all empty stacks that pass through these four points is the intersection of three Voronoi cells: a blue $2$-cell, a green $1$-cell, and an orange $2$-cell. The right panel shows the smallest empty stack in this collection: its center lies on the boundary of the intersection of Voronoi cells, which is the reason why one of its circles passes through an extra point.
• Figure 4: Left: a set of blue points sampled within the indicated three annuli, and another set of orange points sampled outside these annuli so that one blue loop is filled, one half-filled and one empty. Right: the (simplified) barcodes for all points, the blue points, the orange points, and for the kernel of the blue points included into the union.
• Figure 5: The $6$-pack of persistence diagrams for the inclusion of the blue complex into the chromatic complex for the points in Figure \ref{['fig:monkey-set']}. The three arrows indicate the short exact sequences that give rise to the relations between the $1$-norms stated in Theorem \ref{['thm:norm_relations']}.

Theorems & Definitions (7)

• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• Definition 3.4
• Theorem 3.5
• Theorem 4.1
• Theorem 5.1