Related papers: A Lattice for Persistence
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…
The foundational character of certain algebraic structures as Boolean algebras and Heyting algebras is rooted in their potential to model classical and constructive logic, respectively. In this paper we discuss the contributions of…
We provide a naturally isomorphic description of the persistence map from merge trees to barcodes in terms of a monotone map from the partition lattice to the subset lattice. Our description is local, which offers the potential to speed up…
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…
A multicomplex structure is defined from an ordered lattice of multigraphs. This structure will help us to observe the features of Persistent Homology in this context, its interaction with the ordering and the repercussions of the process…
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 redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving…
The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are…
Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph…
Persistent homology enables fast and computable comparison of topological objects. However, it is naturally limited to the analysis of topological spaces. We extend the theory of persistence, by guaranteeing robustness and computability to…
Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and…
Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly…
Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features,…
Solving optimization tasks based on functions and losses with a topological flavor is a very active, growing field of research in data science and Topological Data Analysis, with applications in non-convex optimization, statistics and…
This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence,…
Persistent homology is a technique recently developed in algebraic and computational topology well-suited to analysing structure in complex, high-dimensional data. In this paper, we exposit the theory of persistent homology from first…
In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals…
The machine learning technique of persistent homology classifies complex systems or datasets by computing their topological features over a range of characteristic scales. There is growing interest in applying persistent homology to…
Topological data analysis uses tools from topology -- the mathematical area that studies shapes -- to create representations of data. In particular, in persistent homology, one studies one-parameter families of spaces associated with data,…
Zigzag persistent homology is a powerful generalisation of persistent homology that allows one not only to compute persistence diagrams with less noise and using less memory, but also to use persistence in new fields of application.…