Related papers: Topological Data Analysis in Information Space

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…

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…

In a world abundant with diverse data arising from complex acquisition techniques, there is a growing need for new data analysis methods. In this paper we focus on high-dimensional data that are organized into several hierarchical datasets.…

Topological data analysis is a relatively new branch of machine learning that excels in studying high dimensional data, and is theoretically known to be robust against noise. Meanwhile, data objects with mixed numeric and categorical…

In many areas of applied mathematics, engineering, and social and natural sciences, decentralization of information is a key aspect determining how to approach a problem. In this review article, we study information structures in a…

Diffusion condensation is a dynamic process that yields a sequence of multiscale data representations that aim to encode meaningful abstractions. It has proven effective for manifold learning, denoising, clustering, and visualization of…

This paper lays the foundations for a unified framework for numerically and computationally applying methods drawn from a range of currently distinct geometrical approaches to statistical modelling. In so doing, it extends information…

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…

This paper is devoted to the mathematical study of some divergences based on the mutual information well-suited to categorical random vectors. These divergences are generalizations of the "entropy distance" and "information distance". Their…

Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and…

Topological data analysis is an emerging field that applies the study of topological invariants to data. Perhaps the simplest of these invariants is the number of connected components or clusters. In this work, we explore a topological…

Complex prediction models such as deep learning are the output from fitting machine learning, neural networks, or AI models to a set of training data. These are now standard tools in science. A key challenge with the current generation of…

In many areas of applied geometric/numeric computational mathematics, including geo-mapping, computer vision, computer graphics, finite element analysis, medical imaging, geometric design, and solid modeling, one has to compute incidences,…

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…

Understanding the topological characteristics of data is important to many areas of research. Recent work has demonstrated that synthetic 4D image-type data can be useful to train 4D convolutional neural network models to see topological…

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…

Topological Data Analysis (TDA) uses insights from topology to create representations of data able to capture global and local geometric and topological properties. Its methods have successfully been used to develop estimations of fractal…

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…

Some of the basic concepts of topology are explored through known physics problems. This helps us in two ways, one, in motivating the definitions and the concepts, and two, in showing that topological analysis leads to a clearer…

In recent years, topological data analysis has been utilized for a wide range of problems to deal with high dimensional noisy data. While text representations are often high dimensional and noisy, there are only a few work on the…