Related papers: Persistence weighted Gaussian kernel for topologic…

One of the most important problems arising in time series analysis is that of bifurcation, or change point detection. That is, given a collection of time series over a varying parameter, when has the structure of the underlying dynamical…

We start with a simple introduction to topological data analysis where the most popular tool is called a persistent diagram. Briefly, a persistent diagram is a multiset of points in the plane describing the persistence of topological…

Topological data analysis (TDA) detects geometric structure in biological data. However, many TDA algorithms are memory intensive and impractical for massive datasets. Here, we introduce a statistical protocol that reduces TDA's memory…

Scaling analysis, in which one infers scaling exponents and a scaling function in a scaling law from given data, is a powerful tool for determining universal properties of critical phenomena in many fields of science. However, there are…

Topological Data Analysis (TDA) is a rigorous framework that borrows techniques from geometric and algebraic topology, category theory, and combinatorics in order to study the "shape" of such complex high-dimensional data. Research in this…

We introduce a nonparametric way to estimate the global probability density function for a random persistence diagram. Precisely, a kernel density function centered at a given persistence diagram and a given bandwidth is constructed. Our…

Topological data analysis provides a multiscale description of the geometry and topology of quantitative data. The persistence landscape is a topological summary that can be easily combined with tools from statistics and machine learning.…

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,…

A central problem in data-driven scientific inquiry is how to interpret structure in noisy, high-dimensional data. Topological data analysis (TDA) provides a solution via persistent homology, which encodes features of interest as…

Topological Data Analysis (TDA) has been successfully used for various tasks in signal/image processing, from visualization to supervised/unsupervised classification. Often, topological characteristics are obtained from persistent homology…

Artificial Neural Networks (ANNs) require significant amounts of data and computational resources to achieve high effectiveness in performing the tasks for which they are trained. To reduce resource demands, various techniques, such as…

Persistence diagrams have been widely recognized as a compact descriptor for characterizing multiscale topological features in data. When many datasets are available, statistical features embedded in those persistence diagrams can be…

Developing reliable methods to discriminate different transient brain states that change over time is a key neuroscientific challenge in brain imaging studies. Topological data analysis (TDA), a novel framework based on algebraic topology,…

Machine learning models for repeated measurements are limited. Using topological data analysis (TDA), we present a classifier for repeated measurements which samples from the data space and builds a network graph based on the data topology.…

We show that geometric inference of a point cloud can be calculated by examining its kernel density estimate with a Gaussian kernel. This allows one to consider kernel density estimates, which are robust to spatial noise, subsampling, and…

This thesis consists of two topics related to computational geometry and one topic related to topological data analysis (TDA), which combines fields of computational geometry and algebraic topology for analyzing data. The first part studies…

Understanding the structure of high-dimensional data is fundamental to neuroscience and other data-intensive scientific fields. While persistent homology effectively identifies basic topological features such as "holes," it lacks the…

Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that utilize persistence diagrams in statistical inference, a full Bayesian treatment remains…

Topological Data Analysis (TDA) is a recent approach to analyze data sets from the perspective of their topological structure. Its use for time series data has been limited. In this work, a system developed for a leading provider of cloud…

Persistent homology is a technique recently developed in algebraic and computational topology well-suited to analysing structure in complex, high-dimensional data. In this paper, we exposit the theory of persistent homology from first…