Related papers: $A_\infty$ persistent homology estimates the topol…
Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology.…
Persistent homology captures the evolution of topological features of a model as a parameter changes. The most commonly used summary statistics of persistent homology are the barcode and the persistence diagram. Another summary statistic,…
The alpha complex efficiently computes persistent homology of a point cloud $X$ in Euclidean space when the dimension $d$ is low. Given a subset $A$ of $X$, relative persistent homology can be computed as the persistent homology of the…
Persistent topological properties of an image serve as an additional descriptor providing an insight that might not be discovered by traditional neural networks. The existing research in this area focuses primarily on efficiently…
A topos theoretic generalisation of the category of sets allows for modelling spaces which vary according to time intervals. Persistent homology, or more generally, persistence is a central tool in topological data analysis, which examines…
Persistent homology (PH) is a rigorous mathematical theory that provides a robust descriptor of data in the form of persistence diagrams (PDs) which are 2D multisets of points. Their variable size makes them, however, difficult to combine…
Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a…
A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (in our case, a simplicial complex). Modern packages for persistent homology often construct Vietoris--Rips or other…
We start with a simple introduction to topological data analysis where the most popular tool is called a persistent diagram. Briefly, a persistent diagram is a multiset of points in the plane describing the persistence of topological…
Network centrality measures play a crucial role in understanding graph structures, assessing the importance of nodes, paths, or cycles based on directed or reciprocal interactions encoded by vertices and edges. Estrada and Ross extended…
Observing critical phases in lattice models is challenging due to the need to analyze the finite time or size scaling of observables. We study how the computational topology technique of persistent homology can be used to characterize…
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…
A theory of modules over posets is developed to define computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology with multiple real parameters. To replace…
Extracting useful information from large data sets can be a daunting task. Topological methods for analyzing data sets provide a powerful technique for extracting such information. Persistent homology is a sophisticated tool for identifying…
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…
Multiplexed imaging allows multiple cell types to be simultaneously visualised in a single tissue sample, generating unprecedented amounts of spatially-resolved, biological data. In topological data analysis, persistent homology provides…
In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…
BPS spectra give important insights into the non-perturbative regimes of supersymmetric theories. Often from the study of BPS states one can infer properties of the geometrical or algebraic structures underlying such theories. In this paper…
Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…
Persistent homology is one of the most popular methods in topological data analysis. An initial step in its use involves constructing a nested sequence of simplicial complexes. There is an abundance of different complexes to choose from,…