Related papers: Critical Points to Determine Persistence Homology
In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal…
We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence…
We apply persistent homology, the main method in topological data analysis, to the study of demographic data. Persistence diagrams efficiently summarize information about clusters or peaks in a region's demographic data. To illustrate how…
Critical to the registration of point clouds is the establishment of a set of accurate correspondences between points in 3D space. The correspondence problem is generally addressed by the design of discriminative 3D local descriptors on the…
Persistent homology computes topological invariants from point cloud data. Recent work has focused on developing statistical methods for data analysis in this framework. We show that, in certain models, parametric inference can be performed…
We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and…
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…
Topological complexity is a homotopy invariant that measures the minimal number of continuous rules required for motion planning in a space. In this work, we introduce persistent analogs of topological complexity and its cohomological lower…
The Morse-Smale complex is a standard tool in visual data analysis. The classic definition is based on a continuous view of the gradient of a scalar function where its zeros are the critical points. These points are connected via gradient…
We introduce a new algorithm to parallelise the computation of persistent homology of 2D alpha complexes. Our algorithm distributes the input point cloud among the cores which then compute a cover based on a rectilinear grid. We show how to…
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…
In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…
Processing large point clouds is a challenging task. Therefore, the data is often sampled to a size that can be processed more easily. The question is how to sample the data? A popular sampling technique is Farthest Point Sampling (FPS).…
We study the $k$-th nearest neighbor distance function from a finite point-set in $\mathbb{R}^d$. We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different,…
Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex $K$. These topological changes are summarized in persistence diagrams. We propose…
Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description…
Topological Data Analysis (TDA) has been applied with success to solve problems across many scientific disciplines. However, in the setting of a point cloud $X$ sampled from a shape $\mathcal{S}$ of low intrinsic dimension embedded within…
We develop an analysis pipeline for characterizing the topology of large scale structure and extracting cosmological constraints based on persistent homology. Persistent homology is a technique from topological data analysis that quantifies…