Marco Varisco, Matthew C. B. Zaremsky
We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Vietoris-Rips metric thickenings have previously been proposed as an alternate approach to understanding Vietoris-Rips simplicial complexes and their persistent homology. Recent work has shown that for totally bounded metric spaces, Vietoris-Rips metric thickenings have persistent homology barcodes that agree with those of Vietoris-Rips simplicial complexes, ignoring whether endpoints of bars are open or closed. Combining this result with the known homotopy types and barcodes of the Vietoris-Rips simplicial complexes of the circle, the barcodes of the Vietoris-Rips metric thickenings of the ci…
Motivated by computational aspects of persistent homology for Vietoris-Rips filtrations, we generalize a result of Eliyahu Rips on the contractibility of Vietoris-Rips complexes of geodesic spaces for a suitable parameter depending on the hyperbolicity of the space. We consider the notion of geodesic defect to extend this result to general metric spaces in a way that is also compatible with the filtration. We further show that for finite tree metrics the Vietoris-Rips complexes collapse to their corresponding subforests. We relate our result to modern computational methods by showing that thes…
We construct a compact subset K of the four dimensional Euclidean space with the following property: For all values of the parameter in an interval, the Vietoris-Rips complex of K has uncountably generated first homology. This answers a question that arose in work on persistent homology.
We inspect Vietoris-Rips complexes $VR_t(X)$ of certain metric spaces $X$ using a new generalization of Bestvina-Brady discrete Morse theory. Our main result is a pair of metric criteria on $X$, called the Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain $VR_t(X)$. One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris-Rips complexes. For example we recover some results of Adamaszek-Adams and Hausmann regarding homotopy types of $VR_t(S^n)$. Another application is to geometric gr…
Contractible Vietoris-Rips complexes of $\mathbb{Z}^n$
本文利用 Bestvina-Brady 离散 Morse 理论，为 Virk 关于 $\mathbb{Z}^n$ 的 Vietoris-Rips 复形在足够大尺度下可缩的结果给出了更简短且界更优的新证明，并提出了判定度量空间具有此性质的通用准则。
Multi-parameter persistent homology naturally arises in applications of persistent topology to data that come with extra information depending on additional parameters, like for example time series data. We introduce the concept of a Vietoris-Rips transformation, a method that reduces the computation of the one-parameter persistent homology of pathwise subcomplexes in multi-filtered flag complexes to the computation of the Vietoris-Rips persistent homology of certain semimetric spaces. The corresponding pathwise persistence barcodes track persistence features of the ambient multi-filtered comp…
Persistent homology has emerged as a novel tool for data analysis in the past two decades. However, there are still very few shapes or even manifolds whose persistent homology barcodes (say of the Vietoris-Rips complex) are fully known. Towards this direction, let $P_n$ be the boundary of a regular polygon in the plane with $n$ sides; we describe the homotopy types of Vietoris-Rips complexes of $P_n$. Indeed, when $n=(k+1)!!$ is an odd double factorial, we provide a complete characterization of the homotopy types and persistent homology of the Vietoris-Rips complexes of $P_n$ up to a scale par…
Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes
本文利用 Vietoris-Rips 复形与等变拓扑方法，将不连续函数畸变与 Gromov-Hausdorff 距离联系起来，推广了球面间奇映射的 Borsuk-Ulam 型结果，并改进不同维数球面间 Gromov-Hausdorff 距离的上下界。
A New Construction of the Vietoris-Rips Complex
本文利用$k$-骨架的组合结构提出归纳式Vietoris-Rips复形构造算法，减少单形识别中的比较，在稀疏Erdős-Rényi图和高维复形上较GUDHI v3.9.0实现5–10倍加速。
Let $G$ be a group acting properly and by isometries on a metric space $X$; it follows that the quotient or orbit space $X/G$ is also a metric space. We study the Vietoris-Rips and Čech complexes of $X/G$. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris-Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris-Rips and Čech thickenings of projective spaces a…
On the Classification of Planar-Rips complexes and their corresponding unit disk graphs
本文对可实现为平面点集Vietoris-Rips复形的伪流形与弱伪流形进行单纯同构分类，并同伦分类二维纯闭平面Rips复形，进而刻画一类所有极大团基数相同的单位圆盘图。
Given a sample of points $X$ in a metric space $M$ and a scale $r>0$, the Vietoris-Rips simplicial complex $\mathrm{VR}(X;r)$ is a standard construction to attempt to recover $M$ from $X$ up to homotopy type. A deficiency of this approach is that $\mathrm{VR}(X;r)$ is not metrizable if it is not locally finite, and thus does not recover metric information about $M$. We attempt to remedy this shortcoming by defining a metric space thickening of $X$, which we call the \emph{Vietoris-Rips thickening} $\mathrm{VR}^m(X;r)$, via the theory of optimal transport. When $M$ is a complete Riemannian mani…
The Shadow of Vietoris--Rips Complexes in Limits
本文研究了Vietoris-Rips复形影子在极限下的同伦性质，利用逆系统技术证明极限映射在同伦理论中表现良好，为有限重建问题提供理论支持。
Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applications
该研究针对Vietoris-Rips复形，提出支持局部计算的Forman-Ricci曲率分解及高效局部计算算法，为其几何计算开辟新途径，凸显统计模式支撑几何的重要性。
Vietoris thickenings and complexes of manifolds are homotopy equivalent
该研究证明了维特里斯加厚与流形复形在同伦上的等价性，并展示了其在度量空间中的应用。
On the Vietoris-Rips Complexes of Integer Lattices
本文研究整数格上 Vietoris-Rips 复形的可缩性，证明了 Zaremsky 猜想在 n≤5 时成立，并确定了 r=2 时复形的同伦型。
Given a sample $Y$ from an unknown manifold $X$ embedded in Euclidean space, it is possible to recover the homology groups of $X$ by building a Vietoris--Rips or Čech simplicial complex on top of the vertex set $Y$. However, these simplicial complexes need not inherit the metric structure of the manifold, in particular when $Y$ is infinite. Indeed, a simplicial complex is not even metrizable if it is not locally finite. We instead consider metric thickenings, called the \emph{Vietoris--Rips} and \emph{Čech thickenings}, which are equipped with the 1-Wasserstein metric in place of the simplicia…
We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold $(M,ω)$ with $c_{1}|_{π_{2}(M)}=[ω]|_{π_{2}(M)}=0$. The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently $C^{0}$ close generic function/hamiltonian. The gluing result is a type of Maye…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。