Related papers: On Vietoris-Rips complexes of ellipses

Computing Persistent Homology for large point clouds remains a bottleneck for the wider adoption of persistent homology by the scientific community. We present an algorithm which can compute the degree-1 Vietoris-Rips Persistent Homology of…

In this paper we investigate $m$-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least $m-2$ at any point. These are austere submanifolds in the sense of Harvey and Lawson \cite{harvey} and…

Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces $X$ and $Y$ whenever a map $f:X\to Y$ with strong connectivity conditions on the fibers is given. We apply similar techniques in…

We present a new, inductive construction of the Vietoris-Rips complex, in which we take advantage of a small amount of unexploited combinatorial structure in the $k$-skeleton of the complex in order to avoid unnecessary comparisons when…

We give a new, short proof of a result of Virk, that the Vietoris-Rips complex of the group $\mathbb{Z}^n$ with the standard word metric is contractible at large enough scales. This is inspired by a key observation in Virk's proof, but we…

The Vietoris-Rips filtration for an $n$-point metric space is a sequence of large simplicial complexes adding a topological structure to the otherwise disconnected space. The persistent homology is a key tool in topological data analysis…

We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the…

For X a finite subset of the circle and for 0 < r <= 1 fixed, consider the function f_r : X -> X which maps each point to the clockwise furthest element of X within angular distance less than 2 pi r. We study the discrete dynamical system…

For a finite set of points $V=\{v_1, \dots, v_m\}$ in Euclidean space $\mathbb{R}^d$ and a point $r \in \mathbb{R}^d$, a subset $S \subset V$ is called $r$-balanced if $\mathrm{relint}(\mathrm{conv}(S)) \cap r \neq \emptyset$. In the case…

The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth…

We construct worst-case examples for the standard reduction algorithm for computing persistent homology. Our constructions are similar to the worst-case examples introduced by Morozov, but we replace the single-triangle arrangement with a…

We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree-$k$ in persistent homology, for a either the \cech or the Vietoris--Rips filtration built…

The purpose of this article is to classify the real hypersurfaces in complex space forms of dimension 2 that are both Levi-flat and minimal. The main results are as follows: When the curvature of the complex space form is nonzero, there is…

We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the…

Let $r$ be a positive integer. An $r$-set is a pair $X= (V(X),R(X))$ consisting of a set $V(X)$ with a subset $R(X)$ of the direct product $V(X)^r$. The object of this paper is to investigate the Hom complexes of $r$-sets, which were…

We give bounds for dimension 0 persistent homology and codimension 1 homology of Vietoris--Rips, alpha, and cubical complex filtrations from finite sets related by enrichment (adding new elements), sparsification (removing elements), and…

The presence of outliers can significantly degrade the performance of ellipse fitting methods. We develop an ellipse fitting method that is robust to outliers based on the maximum correntropy criterion with variable center (MCC-VC), where a…

This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of $S^1$) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space…

In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris-Rips, Cech and witness complexes) built on top of precompact spaces. Using recent developments in the theory of topological…

\v{C}ech complexes reveal valuable topological information about point sets at a certain scale in arbitrary dimensions, but the sheer size of these complexes limits their practical impact. While recent work introduced approximation…