Related papers: Mutiscale Mapper: A Framework for Topological Summ…
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…
We present Euler Characteristic Surfaces as a multiscale spatiotemporal topological summary of time series data encapsulating the topology of the system at different time instants and length scales. Euler Characteristic Surfaces with an…
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…
The topology of the large-scale structure of the universe contains valuable information on the underlying cosmological parameters. While persistent homology can extract this topological information, the optimal method for parameter…
We introduce the Stellar decomposition, a model for efficient topological data structures over a broad range of simplicial and cell complexes. A Stellar decomposition of a complex is a collection of regions indexing the complex's vertices…
Complex systems are difficult to study not only because they are nonlinear, multiscale, and often nonstationary, but because their scientifically relevant organization is often invisible at the level of individual components, pairwise…
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…
In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution…
Tabular data is common yet typically incomplete, small in volume, and access-restricted due to privacy concerns. Synthetic data generation offers potential solutions. Many metrics exist for evaluating the quality of synthetic tabular data;…
The Topological Signal Processing (TSP) framework has been recently developed to analyze signals defined over simplicial complexes, i.e. topological spaces represented by finite sets of elements that are closed under inclusion of subsets…
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 an emerging mathematical concept for characterizing shapes in multi-scale data. In this field, persistence diagrams are widely used as a descriptor of the input data, and can distinguish robust and noisy…
One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied…
An understanding of the economic landscape in a world of ever increasing data necessitates representations of data that can inform policy, deepen understanding and guide future research. Topological Data Analysis offers a set of tools which…
In this paper, we develop a neural summarization model which can effectively process multiple input documents and distill Transformer architecture with the ability to encode documents in a hierarchical manner. We represent cross-document…
Given a continuous function $f:X\to\mathbb{R}$ and a cover $\mathcal{I}$ of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover $f^{-1}(\mathcal{I})$. Despite its success in applications, little is known…
With the emergence of graph databases, the task of frequent subgraph discovery has been extensively addressed. Although the proposed approaches in the literature have made this task feasible, the number of discovered frequent subgraphs is…
Topological data analysis (TDA) is a rising field in the intersection of mathematics, statistics, and computer science/data science. The cornerstone of TDA is persistent homology, which produces a summary of topological information called a…
We provide a naturally isomorphic description of the persistence map from merge trees to barcodes in terms of a monotone map from the partition lattice to the subset lattice. Our description is local, which offers the potential to speed up…
One of the primary areas of interest in applied algebraic topology is persistent homology, and, more specifically, the persistence diagram. Persistence diagrams have also become objects of interest in topological data analysis. However,…