Related papers: G-invariant Persistent Homology
Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at…
Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a…
The Gromov--Hausdorff distance (hereinafter referred to as the GH-distance) is a measure of non-isometricity of metric spaces. In this paper, we study a modification of this distance that also takes topological differences into account. The…
We present a novel method to explicitly incorporate topological prior knowledge into deep learning based segmentation, which is, to our knowledge, the first work to do so. Our method uses the concept of persistent homology, a tool from…
Let X be a path-connected topological space admitting a universal cover. Let Homeo(X,a) denote the group of homeomorphisms of X preserving degree one cohomology class a. We investigate the distortion in Homeo(X,a). Let g be an element of…
Topological data analysis (TDA) has emerged as an effective approach in data science, with its key technique, persistent homology, rooted in algebraic topology. Although alternative approaches based on differential topology, geometric…
Persistent homology, while ostensibly measuring changes in topology, captures multiscale geometrical information. It is a natural tool for the analysis of point patterns. In this paper we explore the statistical power of the (persistent…
Fix a finite group $G$ and a conjugacy invariant subset $C\subseteq G$. Let $\Sigma$ be an oriented surface, possibly with punctures. We consider the question of when two homomorphisms $\pi_1(\Sigma) \to G$ taking punctures into $C$ are…
This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using the mathematical concept of topology. We explain the fundamental idea of persistent homology from scratch using some examples. We…
Let $\mathcal{C}$ be a triangulated category with a proper class $\xi$ of triangles. Asadollahi and Salarian introduced and studied $\xi$-Gorenstein projective and $\xi$-Gorenstein injective objects, and developed Gorenstein homological…
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 introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric…
Motivated by geometry processing for surfaces with non-trivial topology, we study discrete harmonic maps between closed surfaces of genus at least two. Harmonic maps provide a natural framework for comparing surfaces by minimizing…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…
This sketch shows that the usual generating function based capacities have an interpretation in the language of persistent homology as persistences of certain homology classes in the persistence module formed by the corresponding generating…
Let $G$ be the classical group, and let Hom$(\mathbb{Z}^m,G)$ denote the space of commuting $m$-tuples in $G$. First, we refine the formula for the Poincar\'e series of Hom$(\mathbb{Z}^m,G)$ due to Ramras and Stafa by assigning (signed)…
One of the main objectives of topological data analysis is the study of discrete invariants for persistence modules, in particular when dealing with multiparameter persistence modules. In many cases, the invariants studied for these…
We propose a novel method for topological analysis of unweighted graphs which is based on \textit{persistent homology}. The proposed method maps the input graph to a complete weighted graph where the weighting function maps each edge to a…
We study linear actions of algebraic groups on smooth projective varieties X. A guiding goal for us is to understand the cohomology of "quotients" under such actions, by generalizing (from reductive to non-reductive group actions) existing…
Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and…