Related papers: Persistence and Topological Complexity
Persistence diagrams are useful displays that give a summary information regarding the topological features of some phenomenon. Usually, only one persistence diagram is available, and replicated persistence diagrams are needed for…
We study the higher (sequential) topological complexity, a numerical homotopy invariant for the planar polygon spaces. For these spaces with a small genetic codes and dimension $m$, Davis showed that their topological complexity is either…
Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a…
This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory.…
We introduce a refinement of persistent homology that detects simple-homotopy-theoretic phenomena invisible to homology. Given a filtered simplicial complex, we define the Morse complexity profile as the minimal number of critical simplices…
Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features (2000) as one varies a tuning parameter. Features with short…
We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the…
Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory. In this…
The topological (resp. geodesic) complexity of a topological (resp. metric) space is roughly the smallest number of continuous rules required to choose paths (resp. shortest paths) between any points of the space. We prove that the geodesic…
In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different,…
Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come.…
We consider persistent homology obtained by applying homology to the open Rips filtration of a compact metric space $(X,d)$. We show that each decrease in zero-dimensional persistence and each increase in one-dimensional persistence is…
In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset $\mathcal{X}$ of a Banach space $\mathbf{Y}$, we analyze the topological features arising in the…
Persistence homology is a tool used to measure topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of…
Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including…
We introduce a variant of Farber's topological complexity, defined for smooth compact orientable Riemannian manifolds, which takes into account only motion planners with the lowest possible "average length" of the output paths. We prove…
We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important…
Latent space matching, which consists of matching distributions of features in latent space, is a crucial component for tasks such as adversarial attacks and defenses, domain adaptation, and generative modelling. Metrics for probability…