Arguably one of the least utilised but most powerful features of persistent homology (PH) is functoriality. Large swathes of data come pre-divided into distinguished, disjoint subsets (a priori/via clustering). A functorial PH pipeline lets us analyse how these subsets interact!

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If you're interested in the topic, here's some recommended reading: Monochromatic Simplicial Collapses Chromatic Delaunay Triangulations Chromatic Alpha Filtrations .

In the monochromatic case, Bauer and Edelsbrunner showed that there are simplicial collapses (and hence homotopy equivalances) Čᵣ(X) ↘ DelČᵣ(X) ↘ 𝒜ᵣ(X) where DelČᵣ(X) is the Delaunay triangulation but with Čech filtration values instead of alpha filtration values.

This filtration is the object of study for my latest pre-print, which is joint work with @nerdarajan, @adambrownphd and @majirouses. Go check it out on the arXiv: Next, I'll summarise what we did and the consequences for practical applications!

Note: - If μ gives a unique colour to each point, we recover the Čech filtration, Č(X). - If μ gives the same colour to every point, we recover the alpha filtration, 𝒜(X). - At every filtration value 𝒜ᵣ(μ) ≃ Čᵣ(X). - When there are few colours, 𝒜(μ) is sparser than Č(X).

Intuitively speaking, this means any map of point clouds, that can be described purely in terms of inclusion of colours, induces an inclusion of subfiltrations. Therefore, we can use PH to study the spatial relationships between points of different colours!

The chromatic alpha filtration (due to di Montesano et al.) provides a trade-off between functoriality and sparsity. For a coloured point cloud μ:X → {0,...,s}, you get a filtration, 𝒜(μ), s.t. given I ⊆ J ⊆ {0,...,s}, there is an inclusion 𝒜(μ⁻¹( I )) ↪ 𝒜(μ⁻¹( J ))

VR has a fatal flaw - there are too many simplices, which slows down PH. For low-dim data, we often turn to the alpha filtration, since it is sparser. However, this filtration is NOT functorial 😔 (removing a point from a Delaunay triangulation does not yield a sub-triangulation)

The Vietoris-Rips pipeline has this property: Given Y⊆X⊆ℝᵈ, there is an inclusion VR(Y) ↪ VR(X). Applying persistent homology, we get a map f: PH(VR(Y)) → PH(VR(X)). Letting Y=🔵, X=🟠∪🔵 in the picture, ker₁(f) detects that the orange points "fill in" the blue points.

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@robertghrist @osumray Thank you! Yes, that tag indicates that the package can produce representatives for the generators. Hopefully we tagged all the right packages. In a future update we could add tool-tips to explain each tag.

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