Related papers: Persistent Entropy for Separating Topological Feat…
Computational topologists recently developed a method, called persistent homology to analyze data presented in terms of similarity or dissimilarity. Indeed, persistent homology studies the evolution of topological features in terms of a…
Topological invariants have played a fundamental role in the advancement of theoretical high energy physics. Physicists have used several kinematic techniques to distinguish new physics predictions from the Standard Model (SM) of particle…
Computation of the simplicial complexes of a large point cloud often relies on extracting a sample, to reduce the associated computational burden. The study considers sampling critical points of a Morse function associated to a point cloud,…
Point-cloud persistent homology (PH) -- computing alpha or Rips complexes on spin-position point clouds -- has been widely applied to detect phase transitions in classical spin models since Donato et al. (2016), with subsequent studies…
We develop a general algebraic framework involving "Poincar\'e--Novikov structures" and "filtered matched pairs" to provide an abstract approach to the barcodes associated to the homologies of interlevel sets of $\mathbb{R}$- or…
Time dependence is a universal phenomenon in nature, and a variety of mathematical models in terms of dynamical systems have been developed to understand the time-dependent behavior of real-world problems. Originally constructed to analyze…
Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram --…
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…
Persistence homology is a tool used to measure topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of…
One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…
Topological data analysis is a recent and fast growing field that approaches the analysis of datasets using techniques from (algebraic) topology. Its main tool, persistent homology (PH), has seen a notable increase in applications in the…
Let $X$ be a closed subspace of a metric space $M$. Under mild hypotheses, one can estimate the Betti numbers of $X$ from a finite set $P \subset M$ of points approximating $X$. In this paper, we show that one can also use $P$ to estimate…
In this paper, we explore the interplay between barcode and topological entropies. Wrapped Floer homology barcode entropy is the exponential growth of not-to-short bars in the persistence module associated with the filtered wrapped Floer…
Persistent homology, an algebraic method for discerning structure in abstract data, relies on the construction of a sequence of nested topological spaces known as a filtration. Two-parameter persistent homology allows the analysis of data…
The paper studies the relation between critical simplices and persistence diagrams of the \v{C}ech filtration. We show that adding a critical $k$-simplex into the filtration corresponds either to a point in the $k$th persistence diagram or…
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 discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point…
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…
Topological data analysis, as a tool for extracting topological features and characterizing geometric shapes, has experienced significant development across diverse fields. Its key mathematical techniques include persistent homology and the…
Persistent homology, the study of holes that appear in data as one thickens balls centered around its points over time, has theoretically guaranteed stability. That is, small data perturbations guarantee small changes in the lifetimes of…