Nerve Models of Subdivision Bifiltrations

We study the size of Sheehy's subdivision bifiltrations, up to homotopy. We focus in particular on the subdivision-Rips bifiltration $\mathcal{SR}(X)$ of a metric space $X$, the only density-sensitive bifiltration on metric spaces known to satisfy a strong robustness property. Given a simplicial filtration $\mathcal{F}$ with a total of $m$ maximal simplices across all indices, we introduce a nerve-based simplicial model for its subdivision bifiltration $\mathcal{SF}$ whose $k$-skeleton has size $O(m^{k+1})$. We also show that the $0$-skeleton of any simplicial model of $\mathcal{SF}$ has size at least $m$. We give several applications: For an arbitrary metric space $X$, we introduce a $\sqrt{2}$-approximation to $\mathcal{SR}(X)$, denoted $\mathcal{J}(X)$, whose $k$-skeleton has size $O(|X|^{k+2})$. This improves on the previous best approximation bound of $\sqrt{3}$, achieved by the degree-Rips bifiltration, which implies that $\mathcal{J}(X)$ is more robust than degree-Rips. Moreover, we show that the approximation factor of $\sqrt{2}$ is tight; in particular, there exists no exact model of $\mathcal{SR}(X)$ with poly-size skeleta. On the other hand, we show that for $X$ in a fixed-dimensional Euclidean space with the $\ell_p$-metric, there exists an exact model of $\mathcal{SR}(X)$ with poly-size skeleta for $p\in \{1, \infty\}$, as well as a $(1+\epsilon)$-approximation to $\mathcal{SR}(X)$ with poly-size skeleta for any $p \in (1, \infty)$ and fixed ${\epsilon > 0}$.