{"title":"细分双层神经模型","authors":"Michael Lesnick, Ken McCabe","doi":"arxiv-2406.07679","DOIUrl":null,"url":null,"abstract":"We study the size of Sheehy's subdivision bifiltrations, up to homotopy. We\nfocus in particular on the subdivision-Rips bifiltration $\\mathcal{SR}(X)$ of a\nmetric space $X$, the only density-sensitive bifiltration on metric spaces\nknown to satisfy a strong robustness property. Given a simplicial filtration\n$\\mathcal{F}$ with a total of $m$ maximal simplices across all indices, we\nintroduce a nerve-based simplicial model for its subdivision bifiltration\n$\\mathcal{SF}$ whose $k$-skeleton has size $O(m^{k+1})$. We also show that the\n$0$-skeleton of any simplicial model of $\\mathcal{SF}$ has size at least $m$.\nWe give several applications: For an arbitrary metric space $X$, we introduce a\n$\\sqrt{2}$-approximation to $\\mathcal{SR}(X)$, denoted $\\mathcal{J}(X)$, whose\n$k$-skeleton has size $O(|X|^{k+2})$. This improves on the previous best\napproximation bound of $\\sqrt{3}$, achieved by the degree-Rips bifiltration,\nwhich implies that $\\mathcal{J}(X)$ is more robust than degree-Rips. Moreover,\nwe show that the approximation factor of $\\sqrt{2}$ is tight; in particular,\nthere exists no exact model of $\\mathcal{SR}(X)$ with poly-size skeleta. On the\nother hand, we show that for $X$ in a fixed-dimensional Euclidean space with\nthe $\\ell_p$-metric, there exists an exact model of $\\mathcal{SR}(X)$ with\npoly-size skeleta for $p\\in \\{1, \\infty\\}$, as well as a\n$(1+\\epsilon)$-approximation to $\\mathcal{SR}(X)$ with poly-size skeleta for\nany $p \\in (1, \\infty)$ and fixed ${\\epsilon > 0}$.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"36 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Nerve Models of Subdivision Bifiltrations\",\"authors\":\"Michael Lesnick, Ken McCabe\",\"doi\":\"arxiv-2406.07679\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the size of Sheehy's subdivision bifiltrations, up to homotopy. We\\nfocus in particular on the subdivision-Rips bifiltration $\\\\mathcal{SR}(X)$ of a\\nmetric space $X$, the only density-sensitive bifiltration on metric spaces\\nknown to satisfy a strong robustness property. Given a simplicial filtration\\n$\\\\mathcal{F}$ with a total of $m$ maximal simplices across all indices, we\\nintroduce a nerve-based simplicial model for its subdivision bifiltration\\n$\\\\mathcal{SF}$ whose $k$-skeleton has size $O(m^{k+1})$. We also show that the\\n$0$-skeleton of any simplicial model of $\\\\mathcal{SF}$ has size at least $m$.\\nWe give several applications: For an arbitrary metric space $X$, we introduce a\\n$\\\\sqrt{2}$-approximation to $\\\\mathcal{SR}(X)$, denoted $\\\\mathcal{J}(X)$, whose\\n$k$-skeleton has size $O(|X|^{k+2})$. This improves on the previous best\\napproximation bound of $\\\\sqrt{3}$, achieved by the degree-Rips bifiltration,\\nwhich implies that $\\\\mathcal{J}(X)$ is more robust than degree-Rips. Moreover,\\nwe show that the approximation factor of $\\\\sqrt{2}$ is tight; in particular,\\nthere exists no exact model of $\\\\mathcal{SR}(X)$ with poly-size skeleta. On the\\nother hand, we show that for $X$ in a fixed-dimensional Euclidean space with\\nthe $\\\\ell_p$-metric, there exists an exact model of $\\\\mathcal{SR}(X)$ with\\npoly-size skeleta for $p\\\\in \\\\{1, \\\\infty\\\\}$, as well as a\\n$(1+\\\\epsilon)$-approximation to $\\\\mathcal{SR}(X)$ with poly-size skeleta for\\nany $p \\\\in (1, \\\\infty)$ and fixed ${\\\\epsilon > 0}$.\",\"PeriodicalId\":501119,\"journal\":{\"name\":\"arXiv - MATH - Algebraic Topology\",\"volume\":\"36 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-06-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Algebraic Topology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2406.07679\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2406.07679","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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 ${\epsilon > 0}$.