Related papers: Spherical Coordinates from Persistent Cohomology
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations…
Persistent homology is a widely-used tool in topological data analysis (TDA) for understanding the underlying shape of complex data. By constructing a filtration of simplicial complexes from data points, it captures topological features…
Manifold-valued signal- and image processing has received attention due to modern image acquisition techniques. Recently, a convex relaxation of the otherwise nonconvex Tikhonov-regularization for denoising circle-valued data has been…
Spherical symmetry arguments are used to produce a general device to convert identities and inequalities for the $p$th absolute moments of real-valued random variables into the corresponding identities and inequalities for the $p$th moments…
Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing…
In this paper, three Computational Topology methods (namely effective homology, persistent homology and discrete vector fields) are mixed together to produce algorithms for homological digital image processing. The algorithms have been…
A central problem in data-driven scientific inquiry is how to interpret structure in noisy, high-dimensional data. Topological data analysis (TDA) provides a solution via persistent homology, which encodes features of interest as…
Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often…
Finding an optimal parameter of a black-box function is important for searching stable material structures and finding optimal neural network structures, and Bayesian optimization algorithms are widely used for the purpose. However, most of…
Persistent homology of the Rips filtration allows to track topological features of a point cloud over scales, and is a foundational tool of topological data analysis. Unfortunately, the Rips-filtration is exponentially sized, when…
Vietoris-Rips metric thickenings have previously been proposed as an alternate approach to understanding Vietoris-Rips simplicial complexes and their persistent homology. Recent work has shown that for totally bounded metric spaces,…
Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as…
A suitable feature representation that can both preserve the data intrinsic information and reduce data complexity and dimensionality is key to the performance of machine learning models. Deeply rooted in algebraic topology, persistent…
Cohomology and cohomology ring of three-dimensional (3D) objects are topological invariants that characterize holes and their relations. Cohomology ring has been traditionally computed on simplicial complexes. Nevertheless, cubical…
Topological data analysis, as a tool for extracting topological features and characterizing geometric shapes, has experienced significant development across diverse fields. Its key mathematical techniques include persistent homology and the…
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…
Recent progress in self-supervised representation learning has resulted in models that are capable of extracting image features that are not only effective at encoding image level, but also pixel-level, semantics. These features have been…
We introduce a new algorithm to parallelise the computation of persistent homology of 2D alpha complexes. Our algorithm distributes the input point cloud among the cores which then compute a cover based on a rectilinear grid. We show how to…
We demonstrate how to use persistent homology for cosmological parameter inference in a tomographic cosmic shear survey. We obtain the first cosmological parameter constraints from persistent homology by applying our method to the…
We present the Chromatic Persistence Algorithm (CPA), an event-driven method for computing persistent cohomological features of weighted graphs via graphic arrangements, a classical object in computational geometry. We establish rigorous…