Related papers: The Higher-Dimensional Skeletonization Problem
Data Science aims to extract meaningful knowledge from unorganised data. Real datasets usually come in the form of a cloud of points with only pairwise distances. Numerous applications require to visualise an overall shape of a noisy cloud…
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 provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…
In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on…
This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of…
Topological methods have the potential of exploring data clouds without making assumptions on their the structure. Here we propose a hierarchical topological clustering algorithm that can be implemented with any distance choice. The…
Geometric graphs form an important family of hidden structures behind data. In this paper, we develop an efficient and robust algorithm to infer a graph skeleton of a high-dimensional point cloud dataset (PCD). Previously, there has been…
Hyperuniformity, the suppression of density fluctuations at large length scales, is observed across a wide variety of domains, from cosmology to condensed matter and biological systems. Although the standard definition of hyperuniformity…
Skeletonization is a powerful tool for shape analysis, rooted in the inherent instinct to understand an object's morphology. It has found applications across various domains, including robotics. Although skeletonization algorithms have been…
Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be…
Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We show that a computational problem closely related to a core task in TDA --…
Persistent homology, a powerful mathematical tool for data analysis, summarizes the shape of data through tracking topological features across changes in different scales. Classical algorithms for persistent homology are often constrained…
This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can…
A method for extracting multiscale geometric features from a data cloud is proposed and analyzed. The basic idea is to map each pair of data points into a real-valued feature function defined on $[0,1]$. The construction of these feature…
Topological data analysis (TDA) is a rapidly developing collection of methods for studying the shape of point cloud and other data types. One popular approach, designed to be robust to noise and outliers, is to first use a smoothing…
The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct…
Topological data analysis is a powerful tool for describing topological signatures in real world data. An important challenge in topological data analysis is matching significant topological signals across distinct systems. In geometry and…
Topological Data Analysis (TDA) can broadly be described as a collection of data analysis methods that find structure in data. This includes: clustering, manifold estimation, nonlinear dimension reduction, mode estimation, ridge estimation…