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  "path": "/abs/2607.06524v1",
  "publishedAt": "2026-07-08T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Kenneth McCabe"
  ],
  "textContent": "**Authors:** Kenneth McCabe\n\nThe 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.",
  "title": "Lower Bounds for Approximating the Vietoris-Rips Filtration"
}