Lower Bounds for Approximating the Vietoris-Rips Filtration

The Vietoris-Rips filtration $\mathcal{VR}(-)$ is a standard tool for analyzing the shape of data within topological data analysis. Beginning with seminal work of Sheehy, a substantial amount of research has centered on constructing linear-size sparse approximations to $\mathcal{VR}(-)$ and related filtrations for metric spaces of bounded doubling dimension. We show that this geometric assumption is necessary in a precise sense. Working in the framework of homotopy interleavings, we show that for any fixed $c \in [1, \sqrt{2})$, there exists a family of finite metric spaces for which any finitely presented $c$-approximation to $\mathcal{VR}(-)$ has exponential size. We also show that for any fixed $c \geq 1$, there exists a family of finite metric spaces for which any finitely presented $c$-approximation to $\mathcal{VR}(-)$ has superlinear size, yielding an obstruction to linear-size approximations for any fixed approximation factor. Both results extend to the intrinsic \v{C}ech filtration and to any bifiltration containing $\mathcal{VR}(-)$ as a $1$-parameter slice, including the function-Rips, degree-Rips, and subdivision-Rips bifiltrations.