相关论文: Field choice problem in persistent homology

In this article we propose a novel approach for comparing the persistent homology representations of two spaces (filtrations). Commonly used methods are based on numerical summaries such as persistence diagrams and persistence landscapes,…

A fundamental challenge in multiparameter persistent homology is the absence of a complete and discrete invariant. To address this issue, we propose an enhanced framework that realizes a holistic understanding of a fully commutative…

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…

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…

Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the…

Recently, persistent homology has had tremendous success in biomolecular data analysis. It works by examining the topological relationship or connectivity of a group of atoms in a molecule at a variety of scales, then rendering a family of…

Persistent (co)homology is a central construction in topological data analysis, where it is used to quantify prominence of features in data to produce stable descriptors suitable for downstream analysis. Persistence is challenging to…

Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances…

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 describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence…

We introduce the persistence heatmap, a parametrized summary based on representative cycles in persistence diagrams, designed to enhance stability and explainability in topological data analysis. Algorithms to compute persistence diagrams…

Persistent Topology studies topological features of shapes by analyzing the lower level sets of suitable functions, called filtering functions, and encoding the arising information in a parameterized version of the Betti numbers, i.e. the…

Persistent homology computes topological invariants from point cloud data. Recent work has focused on developing statistical methods for data analysis in this framework. We show that, in certain models, parametric inference can be performed…

Persistence homology is a tool used to measure topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of…

Long lived topological features are distinguished from short lived ones (considered as topological noise) in simplicial complexes constructed from complex networks. A new topological invariant, persistent homology, is determined and…

Persistent homology computes the multiscale topology of a data set by using a sequence of discrete complexes. In this paper, we propose that persistent homology may be a useful tool for studying the structure of the landscape of string…

Time dependence is a universal phenomenon in nature, and a variety of mathematical models in terms of dynamical systems have been developed to understand the time-dependent behavior of real-world problems. Originally constructed to analyze…

Link prediction is an important learning task for graph-structured data. In this paper, we propose a novel topological approach to characterize interactions between two nodes. Our topological feature, based on the extended persistent…

The concept of topological persistence, introduced recently in computational topology, finds applications in studying a map in relation to the topology of its domain. Since its introduction, it has been extended and generalized in various…

The persistence diagram is a central object in the study of persistent homology and has also been investigated in the context of random topology. The more recent notion of the verbose diagram (a.k.a. verbose barcode) is a refinement of the…