相关论文: Approximate Cech Complexes in Low and High Dimensi…
\v{C}ech complexes are useful simplicial complexes for computing and analyzing topological features of data that lies in Euclidean space. Unfortunately, computing these complexes becomes prohibitively expensive for large-sized data sets…
Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips…
In this paper, we present an algorithm to compute the filtered generalized \v{C}ech complex for a finite collection of disks in the plane, which don't necessarily have the same radius. The key step behind the algorithm is to calculate the…
The \v{C}ech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the \v{C}ech filtration, the number of simplices grows exponentially in the number of input points. A…
Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes constitutes an expensive task because of a combinatorial explosion in the complex size. For $n$ points in…
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…
This paper has three main goals : (1) To give an axiomatic formulation of the construction of "reduced \v{C}ech complexes", complexes using fewer than the usual number of intersections but still computing cohomology of an appropriate class…
In this paper, we present an algorithm that computes the generalized \v{C}ech complex for a finite set of disks where each may have a different radius in 2D space. An extension of this algorithm is also proposed for a set of balls in 3D…
Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes is expensive because of a combinatorial explosion in the complex size. For $n$ points in $\mathbb{R}^d$,…
Fix a finite set of points in Euclidean $n$-space $\euc^n$, thought of as a point-cloud sampling of a certain domain $D\subset\euc^n$. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an…
The so called \v{C}ech and Vietoris-Rips simplicial filtrations are designed to capture information about the topological structure of metric datasets. These filtrations are two of the workhorses in the field of topological data analysis.…
We present a geometric perspective on sparse filtrations used in topological data analysis. This new perspective leads to much simpler proofs, while also being more general, applying equally to Rips filtrations and Cech filtrations for any…
The Vapnik-Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and…
This article introduces proximal Cech nerves and Cech complexes, restricted to finite, bounded regions $K$ of the Euclidean plane. A Cech nerve is a collection of intersecting balls. A Cech complex is a collection of nerves that cover $K$.…
Consider a Poisson point process within a convex set in a Euclidean space. The Vietoris-Rips complex is the clique complex over the graph connecting all pairs of points with distance at most $\delta$. Summing powers of the volume of all…
We introduce a new flavor of functor cocalculus, called \emph{poset cocalculus}, as a tool for studying approximations in topological data analysis. Given a functor from a distributive lattice to a model category, poset cocalculus produces…
For a finite set of balls of radius $r$, the $k$-fold cover is the space covered by at least $k$ balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the $k$-fold filtration of the…
This note proves that only a linear number of holes in a \v{C}ech complex of $n$ points in $\mathbb{R}^d$ can persist over an interval of constant length. Specifically, for any fixed dimension $p < d$ and fixed $\varepsilon > 0$, the number…
Given a set $F$ of $n$ positive functions over a ground set $X$, we consider the problem of computing $x^*$ that minimizes the expression $\sum_{f\in F}f(x)$, over $x\in X$. A typical application is \emph{shape fitting}, where we wish to…
Proximity complexes and filtrations are central constructions in topological data analysis. Built using distance functions, or more generally metrics, they are often used to infer connectivity information from point clouds. Here we…