相关论文: Weighted Persistent Homology

We use topological data analysis to study "functional networks" that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into…

Magnitude homology is an emerging framework that captures the intrinsic topological and geometric features of metric spaces, demonstrating significant potential for topoplogical data analysis and geometric data analysis. This work…

Popular network models such as the mixed membership and standard stochastic block model are known to exhibit distinct geometric structure when embedded into $\mathbb{R}^{d}$ using spectral methods. The resulting point cloud concentrates…

Unravelling the block structure of a network is critical for studying macroscopic features and community-level dynamics. The weighted stochastic block model (WSBM), a variation of the traditional stochastic block model, is designed for…

This article grew out of the theoretical part of my Master's thesis at the Faculty of Mathematics and Information Science at Ruprecht-Karls-Universit\"at Heidelberg under the supervision of PD Dr. Andreas Ott. Following the work of G.…

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…

Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set…

In the theory of persistent homology, a well known duality relates the barcodes of the absolute homology and relative cohomology of a one-parameter simplicial filtration. Motivated by the problem of computing free presentations of the…

0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…

Persistent Homology (PH) is a fundamental tool in computational topology, designed to uncover the intrinsic geometric and topological features of data across multiple scales. Originating within the broader framework of Topological Data…

We propose a unified framework based on persistent homology (PH) to characterize both local and global structures in disordered systems. It can simultaneously generate local and global descriptors using the same algorithm and data…

An algorithm is presented that constructs an acyclic partial matching on the cells of a given simplicial complex from a vector-valued function defined on the vertices and extended to each simplex by taking the least common upper bound of…

Persistent homology is a technique recently developed in algebraic and computational topology well-suited to analysing structure in complex, high-dimensional data. In this paper, we exposit the theory of persistent homology from first…

By their nature it is difficult to differentiate chaotic dynamical systems through measurement. In recent years, work has begun on using methods of Topological Data Analysis (TDA) to qualitatively type dynamical data by approximating the…

Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…

We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent…

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…

While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We study a…

In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals…

We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered…