Related papers: Persistent spectral graph
This paper aims to discuss a method of quantifying the 'shape' of data, via a methodology called topological data analysis. The main tool within topological data analysis is persistent homology; this is a means of measuring the shape of…
Popular network models such as the mixed membership and standard stochastic block model are known to exhibit distinct geometric structure when embedded into $\mathbb{R}^{d}$ using spectral methods. The resulting point cloud concentrates…
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…
In topological data analysis (TDA), one often studies the shape of data by constructing a filtered topological space, whose structure is then examined using persistent homology. However, a single filtered space often does not adequately…
Visualization in the emerging field of topological data analysis has progressed from persistence barcodes and persistence diagrams to display of two-parameter persistent homology. Although persistence barcodes and diagrams have permitted…
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter…
Topological data analysis and its main method, persistent homology, provide a toolkit for computing topological information of high-dimensional and noisy data sets. Kernels for one-parameter persistent homology have been established to…
Persistent Homology is a powerful tool in Topological Data Analysis (TDA) to capture topological properties of data succinctly at different spatial resolutions. For graphical data, shape, and structure of the neighborhood of individual data…
Topological data analysis (TDA) is a rapidly developing collection of methods for studying the shape of point cloud and other data types. One popular approach, designed to be robust to noise and outliers, is to first use a smoothing…
Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex $K$. These topological changes are summarized in persistence diagrams. We propose…
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…
Including intricate topological information (e.g., cycles) provably enhances the expressivity of message-passing graph neural networks (GNNs) beyond the Weisfeiler-Leman (WL) hierarchy. Consequently, Persistent Homology (PH) methods are…
In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals…
We approach the problem of the computation of persistent homology for large datasets by a divide-and-conquer strategy. Dividing the total space into separate but overlapping components, we are able to limit the total memory residency for…
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space.…
This article introduces an algorithm to compute the persistent homology of a filtered complex with various coefficient fields in a single matrix reduction. The algorithm is output-sensitive in the total number of distinct persistent…
Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a…
Persistent homology theory is a relatively new but powerful method in data analysis. Using simplicial complexes, classical persistent homology is able to reveal high dimensional geometric structures of datasets, and represent them as…
Finding an optimal parameter of a black-box function is important for searching stable material structures and finding optimal neural network structures, and Bayesian optimization algorithms are widely used for the purpose. However, most of…
Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such…