Related papers: Computing Optimal Persistent Cycles for Levelset Z…
Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
Over the past two decades, topological data analysis has emerged as a field of applied mathematics with new applications and algorithmic developments appearing rapidly. Two fundamental computations in this field are persistent homology and…
Persistent homology, a powerful mathematical tool for data analysis, summarizes the shape of data through tracking topological features across changes in different scales. Classical algorithms for persistent homology are often constrained…
We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional…
Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the \v{C}ech or Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers.…
We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window (or time-delay) embeddings, as seen through the lens of…
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…
Persistent Homology (PH) offers stable, multi-scale descriptors of intrinsic shape structure by capturing connected components, loops, and voids that persist across scales, providing invariants that complement purely geometric…
This article aims to study the topological invariant properties encoded in node graph representational embeddings by utilizing tools available in persistent homology. Specifically, given a node embedding representation algorithm, we…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the…
We introduce a consistent estimator for the homology (an algebraic structure representing connected components and cycles) of level sets of both density and regression functions. Our method is based on kernel estimation. We apply this…
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…
Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important…
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…
Persistent homology (PH) studies the topology of data across multiple scales by building nested collections of topological spaces called filtrations, computing homology and returning an algebraic object that can be vizualised as a…
As complex networks find applications in a growing range of disciplines, the diversity of naturally occurring and model networks being studied is exploding. The adoption of a well-developed collection of network taxonomies is a natural…