相关论文: Homology Computation of Large Point Clouds using Q…
Structural pattern recognition describes and classifies data based on the relationships of features and parts. Topological invariants, like the Euler number, characterize the structure of objects of any dimension. Cohomology can provide…
Mapper and Ball Mapper are Topological Data Analysis tools used for exploring high dimensional point clouds and visualizing scalar-valued functions on those point clouds. Inspired by open questions in knot theory, new features are added to…
In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
Data analysis in high-dimensional spaces aims at obtaining a synthetic description of a data set, revealing its main structure and its salient features. We here introduce an approach providing this description in the form of a topography of…
Persistent homology is a natural tool for probing the topological characteristics of weighted graphs, essentially focusing on their $0$-dimensional homology. While this area has been substantially studied, we present a new approach to…
Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically -- as long as the algorithm does not…
We present an algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve, i.e., its genus and the properties of the set of fixed points of the anti-holomorphic involution $\tau$,…
The classical method of determining the atomic structure of complex molecules by analyzing diffraction patterns is currently undergoing drastic developments. Modern techniques for producing extremely bright and coherent X-ray lasers allow a…
A methodology is proposed for inferring the topology underlying point cloud data. The approach employs basic elements of Morse Theory, and is capable of producing not only a point estimate of various topological quantities (e.g., genus),…
Hypergraph spectral analysis has emerged as an effective tool processing complex data structures in data analysis. The surface of a three-dimensional (3D) point cloud and the multilateral relationship among their points can be naturally…
With physical quantum computers becoming increasingly accessible, research on their applications across various fields has advanced rapidly. In this paper, we present the first study of quantum cosmology conducted on physical quantum…
By utilizing large-scale graph analytic tools implemented in the modern Big Data platform, Apache Spark, we investigate the topological structure of gravitational clustering in five different universes produced by cosmological $N$-body…
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…
Persistent homology is a method from computational algebraic topology that can be used to study the "shape" of data. We illustrate two filtrations --- the weight rank clique filtration and the Vietoris--Rips (VR) filtration --- that are…
We present a fast algorithm for computing discrete cubical homology of graphs over finite fields with an appropriate characteristic. This algorithm improves on several computational steps compared to constructions in the existing…
Understanding quantum phases and phase transitions in the presence of symmetries is a central objective of quantum many-body physics. A powerful modern paradigm for investigating this problem is topological holography, which relates…
Segmentation is often an essential intermediate step in image analysis. A volume segmentation characterizes the underlying volume image in terms of geometric information--segments, faces between segments, curves in which several faces…
We present an algorithm for quantum-assisted cluster analysis (QACA) that makes use of the topological properties of a D-Wave 2000Q quantum processing unit (QPU). Clustering is a form of unsupervised machine learning, where instances are…
Anisotropic flow measurements in heavy-ion collisions provide important information on the properties of hot and dense matter. These measurements are based on analysis of azimuthal correlations and might be biased by contributions from…