For any fixed c ≥ 1, there exist finite metric spaces whose Vietoris-Rips filtration cannot be c-approximated by any finitely presented construction of linear size; for c < √2, exponential size is required.
Computing the multicover bifiltration.Discrete & Computational Geometry, 70(2):376– 405, 2023
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Lower Bounds for Approximating the Vietoris-Rips Filtration
For any fixed c ≥ 1, there exist finite metric spaces whose Vietoris-Rips filtration cannot be c-approximated by any finitely presented construction of linear size; for c < √2, exponential size is required.