Related papers: Tropical Coordinates on the Space of Persistence B…
This short paper presents a general approach for computing robust Wasserstein barycenters of persistence diagrams. The classical method consists in computing assignment arithmetic means after finding the optimal transport plans between the…
Topological Data Analysis (TDA) is a rising field of computational topology in which the topological structure of a data set can be observed by persistent homology. By considering a sequence of sublevel sets, one obtains a filtration that…
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…
Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…
Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram --…
Consider the space of continuous functions on a geometric tree $X$ whose persistent homology gives rise to a finite generic barcode $D$. We show that there are exactly as many path connected components in this space as there are merge trees…
Delay-coordinate mapping is an effective and widely used technique for reconstructing and analyzing the dynamics of a nonlinear system based on time-series outputs. The efficacy of delay-coordinate mapping has long been supported by Takens'…
Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certainfiltering function.…
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…
We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as…
As the field of Topological Data Analysis continues to show success in theory and in applications, there has been increasing interest in using tools from this field with methods for machine learning. Using persistent homology, specifically…
The $\gamma$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules…
While there has been much interest in adapting conventional clustering procedures---and in higher dimensions, persistent homology methods---to directed networks, little is known about the convergence of such methods. In order to even…
Robust topological information commonly comes in the form of a set of persistence diagrams, finite measures that are in nature uneasy to affix to generic machine learning frameworks. We introduce a fast, learnt, unsupervised vectorization…
A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning…
We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale…
This paper surveys {\it tropical modifications}, which have already become a folklore in tropical geometry. Tropical modifications are used in tropical intersection theory, tropical Hodge theory, and in the study of singularities. They…
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…
Stickiness is a well known phenomenon in which chaotic orbits expend an expressive amount of time in specific regions of the chaotic sea. This phenomenon becomes important when dealing with area-preserving open systems because, in this…
In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction…