Related papers: Geometry of Data
Scientists use a mathematical subject called 'topology' to study the shapes of objects. An important part of topology is counting the numbers of pieces and holes in objects, and people use this information to group objects into different…
Topological data analysis provides a collection of tools to encapsulate and summarize the shape of data. Currently it is mainly restricted to \emph{mapper algorithm} and \emph{persistent homology}. In this paper we introduce new…
Topological data analysis refers to approaches for systematically and reliably computing abstract ``shapes'' of complex data sets. There are various applications of topological data analysis in life and data sciences, with growing interest…
We use the notion of topological data analysis to compare metrics on data sets. We provide two different motivating examples for this. The first of these is a point cloud data set that has $\mathbb{R}^2$ as its ambient space, and is…
Homology-based invariants can be used to characterize the geometry of datasets and thereby gain some understanding of the processes generating those datasets. In this work we investigate how the geometry of a dataset changes when it is…
A short survey on applications of algebraic geometry in topological data analysis.
Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a…
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…
We perform topological data analysis on the internal states of convolutional deep neural networks to develop an understanding of the computations that they perform. We apply this understanding to modify the computations so as to (a) speed…
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.…
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…
We develop a new concept of non-positive curvature for metric spaces, based on intersection patterns of closed balls. In contrast to the synthetic approaches of Alexandrov and Buesemann, our concept also applies to metric spaces that might…
Finance is heavily influenced by data-driven decision-making. Meanwhile, our ability to comprehend the full informational content of data sets remains impeded by the tools we apply in analysis, especially where the data is high-dimensional.…
A primary hypothesis that drives scientific and engineering studies is that data has structure. The dominant paradigms for describing such structure are statistics (e.g., moments, correlation functions) and signal processing (e.g.,…
Topological Data Analysis (TDA) can broadly be described as a collection of data analysis methods that find structure in data. This includes: clustering, manifold estimation, nonlinear dimension reduction, mode estimation, ridge estimation…
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…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be…
While the strength of Topological Data Analysis has been explored in many studies on high dimensional numeric data, it is still a challenging task to apply it to text. As the primary goal in topological data analysis is to define and…
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,…