Related papers: Stable Volumes for Persistent Homology
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…
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…
Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such…
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…
We use methods from topological data analysis to study the topological features of certain distributions of string vacua. Topological data analysis is a multi-scale approach used to analyze the topological features of a dataset by…
Persistent homology theory is a relatively new but powerful method in data analysis. Using simplicial complexes, classical persistent homology is able to reveal high dimensional geometric structures of datasets, and represent them as…
Classical Finite Volume methods for multi-dimensional problems include stabilization (e.g.\ via a Riemann solver), that is derived by considering several one-dimensional problems in different directions. Such methods therefore ignore a…
Persistent Homology (PH) is a fundamental tool in computational topology, designed to uncover the intrinsic geometric and topological features of data across multiple scales. Originating within the broader framework of Topological Data…
We introduce Cubical Ripser for computing persistent homology of image and volume data (more precisely, weighted cubical complexes). To our best knowledge, Cubical Ripser is currently the fastest and the most memory-efficient program for…
Inspired by topological data analysis techniques, we introduce persistent homology observables and apply them in a geometric analysis of the dynamics of quantum field theories. As a prototype application, we consider data from a…
Persistent homology is a popular technique in topological data analysis that tracks the lifespans of homological features in a nested sequence of spaces. This data is typically presented in a multi-set called a persistence diagram or a…
Stable quantum computation requires noisy results to remain bounded even in the presence of noise fluctuations. Yet non-stationary noise processes lead to drift in the varying characteristics of a quantum device that can greatly influence…
In recent years, topological data analysis has been utilized for a wide range of problems to deal with high dimensional noisy data. While text representations are often high dimensional and noisy, there are only a few work on the…
Single-parameter persistent homology, a key tool in topological data analysis, has been widely applied to data problems along with statistical techniques that quantify the significance of the results. In contrast, statistical techniques for…
We consider the degree-Rips construction from topological data analysis, which provides a density-sensitive, multiparameter hierarchical clustering algorithm. We analyze its stability to perturbations of the input data using the…
One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…
Topological methods, including persistent homology, are powerful tools for analysis of high-dimensional data sets but these methods rely almost exclusively on thresholding techniques. In noisy data sets, thresholding does not always allow…
Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates,…
We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method…
Persistent (co)homology is a central construction in topological data analysis, where it is used to quantify prominence of features in data to produce stable descriptors suitable for downstream analysis. Persistence is challenging to…