Abhinav Natarajan

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!

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)

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!