Related papers: Demystifying Latschev's Theorem: Manifold Reconstr…

For a sufficiently small scale $\beta>0$, the Vietoris$\unicode{x2013}$Rips complex $\mathcal{R}_\beta(S)$ of a metric space $S$ with a small Gromov$\unicode{x2013}$Hausdorff distance to a closed Riemannian manifold $M$ has been already…

Latschev's theorem provides sufficient conditions on a metric space $M$ and $\delta > 0$ for the homotopy type of $M$ to agree with that of the Vietoris-Rips complex $\mathcal{R}^{\delta}(N)$ of any nearby space $N$ in the Gromov-Hausdorff…

Selective Rips complexes corresponding to a sequence of parameters are a generalization of Vietoris-Rips complexes utilizing the idea of thin simplices. We prove that if a metric space $Y$ is close (in Gromov-Hausdorff distance) to a closed…

We consider the problem of homotopy-type reconstruction of compact subsets $X\subset\R^N$ that have the Alexandrov curvature bounded above ($\leq$ $\kappa$) in the intrinsic length metric. The reconstructed spaces are in the form of…

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 \v{C}ech simplicial complex on top of the vertex set $Y$. However, these…

We derive conditions under which the reconstruction of a target space is topologically correct via the \v{C}ech complex or the Vietoris-Rips complex obtained from possibly noisy point cloud data. We provide two novel theoretical results.…

We consider the topological and geometric reconstruction of a geodesic subspace of $\mathbb{R}^N$ both from the \v{C}ech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages…

The shadow of an abstract simplicial complex $K$ with vertices in $\mathbb{R}^N$ is a subset of $\mathbb{R}^N$ defined as the union of the convex hulls of simplices of $K$. The Vietoris--Rips complex of a metric space $(S,d)$ at scale…

Given a metric space X and a distance threshold r>0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and…

Inspired by a recent work of Wang-Zhao, in this note we prove that for a fixed $n$-dimensional closed Riemannian manifold $(M^n, g)$, if an $\mathrm{RCD}(K, n)$ space $(X, \mathsf{d}, \mathfrak{m})$ is Gromov-Hausdorff close to $M^n$, then…

Often noisy point clouds are given as an approximation of a particular compact set of interest. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the…

Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a…

We develop novel methods for using persistent homology to infer the homology of an unknown Riemannian manifold $(M, g)$ from a point cloud sampled from an arbitrary smooth probability density function. Standard distance-based filtered…

We show that if $X$ is a finite-dimensional Polish metric space, then the natural bijection $\mathrm{VR}(X;r)\to \mathrm{VR^m}(X;r)$ from the (open) Vietoris-Rips complex to the Vietoris-Rips metric thickening is a homotopy equivalence.…

Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes…

From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopy-equivalent to that manifold with high confidence. We describe a corresponding…

Let $M$ be a closed symplectic manifold of dimension $2n$ with non-ellipticity. We can define an almost K\"ahler structure on $M$ by using the given symplectic form. Hence, we have a $\G=\pi_1(M)$-invariant almost K\"ahler structure on the…

We study the concepts of the $\ell_p$-Vietoris-Rips simplicial set and the $\ell_p$-Vietoris-Rips complex of a metric space, where $1\leq p \leq \infty.$ This theory unifies two established theories: for $p=\infty,$ this is the classical…

Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded, concave, connected open set $U$ with $C^0$ boundary and $M\setminus U$ is connected. We assume that either the relative homotopy set $\pi_1(M,M\setminus U)=0$ or the…

Manifold fitting aims to reconstruct a low-dimensional manifold from high-dimensional data, whose framework is established by Fefferman et al. \cite{fefferman2020reconstruction,fefferman2021reconstruction}. This paper studies the recovery…