Thomas Chaplin

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!

I'm super excited to have a new preprint - Manifold Diffusion Geometry Diffusion geometry *robustly* measures curvature, dimension, and tangent spaces of data! arxiv.org/abs/2411.04100

1/5 Very happy to share our work on flexible, multiscale cell type assignment for subcellular spatial transcriptomics, now in @Nature nature.com/articles/s4158…

If you're interested in the topic, here's some recommended reading: Monochromatic Simplicial Collapses dx.doi.org/10.1090/tran/6… Chromatic Delaunay Triangulations arxiv.org/abs/2212.03121 Chromatic Alpha Filtrations arxiv.org/abs/2212.03128 .

If you find this at all interesting, please checkout the pre-print: arxiv.org/abs/2405.19303 or ask any questions here! We've also released a Python library that can compute these filtrations: abhinavnatarajan.github.io/Chalc/ Finally, a big thank you to my wonderful collaborators!

We extend this result to the coloured case. Our proof is very much an adptation of their's to the chromatic case. Moreover, we show that if ν refines μ then DelČᵣ(ν) ↘ DelČᵣ(μ). This is achieved via a novel Morse function, that is compatible with the Čech filtration function.

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.

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!

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: arxiv.org/abs/2405.19303 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.

Wonderful week of rainbows, maths and manuscripts! Chromatic work by Abhinav Natarajan @tomrchaplin @majirouses @adambrownphd Toric fibre products and MLE by Niharika Paul @JaneIvyCoons @tudresden_de @OxUniMaths arxiv.org/abs/2405.19303 arxiv.org/abs/2405.13897

Baby #2 is due any day now, and I am too wired to do the intelligent thing (i.e., sleep). So please join me, the wonderful @osumray, and @haharrington on a journey through some fun machine learning & representation theory. 1/11

Ever wondered how music recognition apps like Shazam work or why they fail? Can Algebraic Topology improve in audio ID algorithms? Our new work in collaboration with @Spotify, 'Topological Fingerprints for Audio ID', arxiv.org/pdf/2309.03516… dives deep into this problem. [1/n]

@prof_g @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.

nice job! i'm a little confused as to what is meant by the tag type/representative. does that mean it gives representatives of explicit generators? if not, can you add a tag that indicates whether the package gives you explicit generators in the original data?

Starting a new TDA project? Need persistent homology on cubical complexes and a custom filtration in C++? Or just a Rips filtration but in parallel and using Python? Wading through all those repos giving you a headache? 1/3