Related papers: An introduction to Topological Data Analysis: fund…

Visualization of data is an important step in the understanding of data and the evaluation of statistical models. Topological Data Analysis Ball Mapper (TDABM) after Dlotko (2019), provides a model free means to visualize multivariate…

This article is a short survey of the history of functional analysis for topologists.

Topological Data Analysis (TDA) has emerged as a powerful tool for extracting meaningful features from complex data structures, driving significant advancements in fields such as neuroscience, biology, machine learning, and financial…

As new technologies move to the fore, our understanding of the world may seem to have shrunk in comparison, for despite new developments in research, much of it is reduced or rather, abstracted for marketability. Thus, the purpose of this…

We live in a world where data generation is omnipresent. Innovations in computer hardware in the last few decades coupled with increasingly reliable connectivity among them have fueled this phenomenon. We are constantly creating and…

This paper contains an expository account of persistent homology and its usefulness for topological data analysis. An alternative foundation for level-set persistence is presented using sheaves and cosheaves.

The continuous increase of data generated provides enormous possibilities of both public and private companies. The management of this mass of data or big data will play a crucial role in the society of the future, as it finds applications…

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data --- for example, to approximate a point cloud by a low-dimensional…

Data visualization is the process by which data of any size or dimensionality is processed to produce an understandable set of data in a lower dimensionality, allowing it to be manipulated and understood more easily by people. The goal of…

In the 1990s, statisticians began thinking in a principled way about how computation could better support the learning and doing of statistics. Since then, the pace of software development has accelerated, advancements in computing and data…

Recent development of network structure analysis shows that it plays an important role in characterizing complex system of many branches of sciences. Different from previous network centrality measures, this paper proposes the notion of…

Object oriented data analysis is the statistical analysis of populations of complex objects. In the special case of functional data analysis, these data objects are curves, where standard Euclidean approaches, such as principal component…

This is a brief review, in relatively non-technical terms, of recent advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric…

Big data is transforming our world, revolutionizing operations and analytics everywhere, from financial engineering to biomedical sciences. The complexity of big data often makes dimension reduction techniques necessary before conducting…

Information Theory provides a fundamental basis for analysis, and for a variety of subsequent methodological approaches, in relation to uncertainty quantification. The transversal character of concepts and derived results justifies its…

Advances in imaging techniques enable high resolution 3D visualisation of vascular networks over time and reveal abnormal structural features such as twists and loops, and their quantification is an active area of research. Here we showcase…

Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and…

Topological data analysis is a powerful tool for describing topological signatures in real world data. An important challenge in topological data analysis is matching significant topological signals across distinct systems. In geometry and…

Scientific data has been growing in both size and complexity across the modern physical, engineering, life and social sciences. Spatial structure, for example, is a hallmark of many of the most important real-world complex systems, but its…

As both machine learning models and the datasets on which they are evaluated have grown in size and complexity, the practice of using a few summary statistics to understand model performance has become increasingly problematic. This is…