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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…
Over the last two decades, topological data analysis (TDA) has emerged as a very powerful data analytic approach which can deal with various data modalities of varying complexities. One of the most commonly used tools in TDA is persistent…
This survey provides a comprehensive exploration of applications of Topological Data Analysis (TDA) within neural network analysis. Using TDA tools such as persistent homology and Mapper, we delve into the intricate structures and behaviors…
This article aims to study the topological invariant properties encoded in node graph representational embeddings by utilizing tools available in persistent homology. Specifically, given a node embedding representation algorithm, we…
Topology identification and inference of processes evolving over graphs arise in timely applications involving brain, transportation, financial, power, as well as social and information networks. This chapter provides an overview of graph…
In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately…
Topological Data Analysis (TDA) has been praised by researchers for its ability to capture intricate shapes and structures within data. TDA is considered robust in handling noisy and high-dimensional datasets, and its interpretability is…
Surface roughness plays an important role in analyzing engineering surfaces. It quantifies the surface topography and can be used to determine whether the resulting surface finish is acceptable or not. Nevertheless, while several existing…
Topological data analysis involves the statistical characterization of the shape of data. Persistent homology is a primary tool of topological data analysis, which can be used to analyze topological features and perform statistical…
Topological data analysis (TDA) studies the shape patterns of data. Persistent homology is a widely used method in TDA that summarizes homological features of data at multiple scales and stores them in persistence diagrams (PDs). In this…
Topological data analysis (TDA) provides insight into data shape. The summaries obtained by these methods are principled global descriptions of multi-dimensional data whilst exhibiting stable properties such as robustness to deformation and…
In this work we use the persistent homology method, a technique in topological data analysis (TDA), to extract essential topological features from the data space and combine them with deep learning features for classification tasks. In TDA,…
Natural data offer a hard challenge to data analysis. One set of tools is being developed by several teams to face this difficult task: Persistent topology. After a brief introduction to this theory, some applications to the analysis and…
Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features (2000) as one varies a tuning parameter. Features with short…
Topological data analysis (TDA) is an active field of mathematics for quantifying shape in complex data. Standard methods in TDA such as persistent homology (PH) are typically focused on the analysis of data consisting of a single entity…
Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram --…
Topological Data Analysis (TDA) is a field that leverages tools and ideas from algebraic topology to provide robust methods for analysing geometric and topological aspects of data. One of the principal tools of TDA, persistent homology,…
Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of 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…
Identifying molecular signatures from complex disease patients with underlying symptomatic similarities is a significant challenge in the analysis of high dimensional multi-omics data. Topological data analysis (TDA) provides a way of…