Related papers: DTM-based Filtrations
As a popular representation of 3D data, point cloud may contain noise and need to be filtered before use. Existing point cloud filtering methods either cannot preserve sharp features or result in uneven point distribution in the filtered…
The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used…
This paper investigates differentially private analysis of distance-based outliers. The problem of outlier detection is to find a small number of instances that are apparently distant from the remaining instances. On the other hand, the…
This work incorporates topological features via persistence diagrams to classify point cloud data arising from materials science. Persistence diagrams are multisets summarizing the connectedness and holes of given data. A new distance on…
Topological Data Analysis (TDA) is an emergent field that aims to discover topological information hidden in a dataset. TDA tools have been commonly used to create filters and topological descriptors to improve Machine Learning (ML)…
Given a set P of n points and a constant k, we are interested in computing the persistent homology of the Cech filtration of P for the k-distance, and investigate the effectiveness of dimensionality reduction for this problem, answering an…
Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data --- for example, to approximate a point cloud by a low-dimensional…
Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams…
We study detection of collapse in high-dimensional point clouds, where mass concentrates near a lower-dimensional set relative to a non-collapsed geometry. We propose persistent homology-based test statistics under two well-studied…
Tools of Topological Data Analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well studied data summary, suffers a number of limitations; its computations are hard to…
Directed graphs arise in many applications where computing persistent homology helps to encode the shape and structure of the input information. However, there are only a few ways to turn the directed graph information into an undirected…
This paper is the first to attempt differentially private (DP) topological data analysis (TDA), producing near-optimal private persistence diagrams. We analyze the sensitivity of persistence diagrams in terms of the bottleneck distance, and…
The combination of the Internet of Things and the Edge Computing gives many opportunities to support innovative applications close to end users. Numerous devices present in both infrastructures can collect data upon which various processing…
Persistent homology studies the evolution of k-dimensional holes along a nested sequence of simplicial complexes (called a filtration). The set of bars (i.e. intervals) representing birth and death times of k-dimensional holes along such…
In this paper we define, implement, and investigate a simplicial complex construction for computing persistent homology of Euclidean point cloud data, which we call the Delaunay-Rips complex (DR). Assigning the Vietoris-Rips weights to…
Persistent homology, an algebraic method for discerning structure in abstract data, relies on the construction of a sequence of nested topological spaces known as a filtration. Two-parameter persistent homology allows the analysis of data…
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and…
We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with…
Removing outlier correspondences is one of the critical steps for successful feature-based point cloud registration. Despite the increasing popularity of introducing deep learning methods in this field, spatial consistency, which is…
In this paper, we present a robust spherical harmonics approach for the classification of point cloud-based objects. Spherical harmonics have been used for classification over the years, with several frameworks existing in the literature.…