🤖 AI Summary
This work investigates the fundamental limits of sparsely approximating Vietoris–Rips filtrations without any geometric assumptions. By constructing specific families of metric spaces and combining techniques from homotopy interleaving theory, combinatorial topology, and metric geometry, the authors establish the first lower bounds on approximation size within the homotopy interleaving framework: for any fixed approximation factor \( c \in [1, \sqrt{2}) \), there exist instances requiring exponentially large representations, and for any \( c \geq 1 \), the approximation size must be super-linear. These results demonstrate that geometric conditions—such as bounded doubling dimension—are essential for achieving efficient approximations and extend to several bifiltration settings, thereby ruling out the possibility of universally linear-size approximations.
📝 Abstract
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 Čech filtration and to any bifiltration containing $\mathcal{VR}(-)$ as a $1$-parameter slice, including the function-Rips, degree-Rips, and subdivision-Rips bifiltrations.