相关论文: The (homological) persistence of gerrymandering

Gerrymandering is one of the biggest threats to American democracy. By manipulating district lines, politicians effectively choose their voters rather than the other way around. Current gerrymandering identification methods (namely the…

We apply persistent homology, the main method in topological data analysis, to the study of demographic data. Persistence diagrams efficiently summarize information about clusters or peaks in a region's demographic data. To illustrate how…

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…

In topological data analysis, persistent homology characterizes robust topological features in data and it has a summary representation, called a persistence diagram. Statistical research for persistence diagrams have been actively…

It is important to choose the geographical distributions of public resources in a fair and equitable manner. However, it is complicated to quantify the equity of such a distribution; important factors include distances to resource sites,…

Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph…

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…

Topological Data Analysis (TDA) is a novel statistical technique, particularly powerful for the analysis of large and high dimensional data sets. Much of TDA is based on the tool of persistent homology, represented visually via persistence…

Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly…

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,…

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…

Persistent homology is a tool from Topological Data Analysis (TDA) used to summarize the topology underlying data. It can be conveniently represented through persistence diagrams. Observing a noisy signal, common strategies to infer its…

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…

Deciding whether a political districting plan was distorted by a hidden agenda, or whether it dilutes the voting power of some group, requires a neutral baseline for comparison. Remarkably, all nine U.S. Supreme Court justices have now…

In this paper we develop a novel Topological Data Analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a…

Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams…

Persistent homology is an important methodology in topological data analysis which adapts theory from algebraic topology to data settings. Computing persistent homology produces persistence diagrams, which have been successfully used in…

Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…

Graph Neural Networks (GNNs) have become the standard for graph representation learning but remain vulnerable to structural perturbations. We propose a novel framework that integrates persistent homology features with stability…

This paper is a survey of persistent homology, primarily as it is used in topological data analysis. It includes the theory of persistence modules, as well as stability theorems for persistence barcodes, generalized persistence,…