Related papers: Computing the Bottleneck Distance between Persiste…
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…
To analyze the topological properties of the given discrete data, one needs to consider a continuous transform called filtration. Persistent homology serves as a tool to track changes of homology in the filtration. The outcome of the…
The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen-Steiner, Edelsbrunner, and Harer. We…
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.…
The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the $\infty$-norm of…
Persistent homology is a tool of topological data analysis that has been used in a variety of settings to characterize different dimensional holes in data. However, persistent homology computations can be memory intensive with a…
Directed graphs arise in many applications where computing persistent homology helps to encode the shape and structure of the input information. However, there are only a few ways to turn the directed graph information into an undirected…
Persistent homology provides information about the lifetime of homology classes along a filtration of cell complexes. Persistence barcode is a graphical representation of such information. A filtration might be determined by time in a set…
Optimal transport (OT) provides powerful tools for comparing probability measures in various types. The Wasserstein distance which arises naturally from the idea of OT is widely used in many machine learning applications. Unfortunately,…
In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…
The field of mathematical morphology offers well-studied techniques for image processing. In this work, we view morphological operations through the lens of persistent homology, a tool at the heart of the field of topological data analysis.…
We apply persistent homology to the task of discovering and characterizing phase transitions, using lattice spin models from statistical physics for working examples. Persistence images provide a useful representation of the homological…
We present a fast algorithm for global rigid symmetry detection with approximation guarantees. The algorithm is guaranteed to find the best approximate symmetry of a given shape, to within a user-specified threshold, with very high…
The persistence barcode is a topological descriptor of data that plays a fundamental role in topological data analysis. Given a filtration of the space of data, a persistence barcode tracks the evolution of its homological features. In this…
Capturing the dynamics of active particles, i.e., small self-propelled agents that both deform and are deformed by a fluid in which they move is a formidable problem as it requires coupling fine scale hydrodynamics with large scale…
Features such as photon rings, jets, or hot. spots can leave particular topological signatures in a black hole image. As such, topological data analysis can be used to characterize images resulting from high resolution observations…
In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on…
Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of…
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,…
In this study, we introduce novel methodologies designed to adapt original data in response to the dynamics of persistence diagrams along Wasserstein gradient flows. Our research focuses on the development of algorithms that translate…