Related papers: An Introduction to Multiparameter Persistence
Topological Data Analysis (TDA) provides powerful tools to explore the shape and structure of data through topological features such as clusters, loops, and voids. Persistence diagrams are a cornerstone of TDA, capturing the evolution of…
Persistent homology is a widely-used tool in topological data analysis (TDA) for understanding the underlying shape of complex data. By constructing a filtration of simplicial complexes from data points, it captures topological features…
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…
In this chapter, we discuss applications of topological data analysis (TDA) to spatial systems. We briefly review the recently proposed level-set construction of filtered simplicial complexes, and we then examine persistent homology in two…
Persistent homology is a powerful tool for characterizing the topology of a data set at various geometric scales. When applied to the description of molecular structures, persistent homology can capture the multiscale geometric features and…
Carlsson, Singh and Memoli's TDA mapper takes a point cloud dataset and outputs a graph that depends on several parameter choices. Dey, Memoli, and Wang developed Multiscale Mapper for abstract topological spaces so that parameter choices…
The study of topology is strictly speaking, a topic in pure mathematics. However in only a few years, Topological Data Analysis (TDA), which refers to methods of utilizing topological features in data (such as connected components, tunnels,…
The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has…
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…
The Mapper algorithm, a technique within topological data analysis (TDA), constructs a simplified graphical representation of high-dimensional data to uncover its underlying shape and structural patterns. The algorithm has attracted…
Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding…
Multiparameter persistent homology is a generalization of classical persistent homology, a central and widely-used methodology from topological data analysis, which takes into account density estimation and is an effective tool for data…
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…
The surge of data available on the Internet has driven the adoption of a wide range of computational methods for analyzing and extracting insights from large-scale data. Among these, Machine Learning (ML) has become a central paradigm,…
One of the main challenges of Topological Data Analysis (TDA) is to extract features from persistent diagrams directly usable by machine learning algorithms. Indeed, persistence diagrams are intrinsically (multi-)sets of points in…
Understanding the behavior and evolution of a dynamical many-body system by analyzing patterns in their experimentally captured images is a promising method relevant for a variety of living and non-living self-assembled systems. The arrays…
Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in…
Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using persistence landscapes…
Inferring topological and geometrical information from data can offer an alternative perspective on machine learning problems. Methods from topological data analysis, e.g., persistent homology, enable us to obtain such information,…
Spatial transcriptomics studies are becoming increasingly large and commonplace, necessitating simultaneous analysis of a large number of spatially resolved variables. Correspondingly, a diverse range of methodologies have been proposed to…