Lower Bounds for Approximating the Vietoris-Rips Filtration

What the math gives to ML

The paper provides a constructive certificate for when a filtration cannot be compressed below a specified complexity: the rank of a persistent-homology structure map forces every finitely presented approximation to retain enough independent topological features. This can be transferred to topology-aware token, point, or edge pruning in geometric neural networks, where the rank between two scales becomes a data-dependent lower bound on the number of latent slots or representatives that may safely be retained. The strongest practical use is not to reproduce a Vietoris–Rips approximation exactly, but to use persistent rank as a stopping rule that prevents aggressive compression from destroying multiscale cycles and connected components.