Introduction to Persistent Homology

This video presents an introduction to persistent homology, aimed at a viewer who has mathematical aptitude but not necessarily knowledge of algebraic topology. Persistent homology is an algebraic method of discerning the topological features of complex data, which in recent years has found applications in fields such as image processing and biological systems. Using smooth animations, the video conveys the intuition behind persistent homology, while giving a taste of its key properties, applications, and mathematical underpinnings

Stratifying multiparameter persistent homology

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 family of spaces obtained from the data. In applications data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well-understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.Comment: Minor improvements throughout. In particular: we extended the introduction, added Table 1, which gives a dictionary between terms used in PH and commutative algebra; we streamlined Section 3; we added Proposition 4.49 about the information captured by the cp-rank; we moved the code from the appendix to github. Final version, to appear in SIAG

Introduction to Computational Topology Using Simplicial Persistent Homology

The human mind has a natural talent for finding patterns and shapes in nature where there are none, such as constellations among the stars. Persistent homology serves as a mathematical tool for accomplishing the same task in a more formal setting, taking in a cloud of individual points and assembling them into a coherent continuous image. We present an introduction to computational topology as well as persistent homology, and use them to analyze configurations of BuckyBalls®, small magnetic balls commonly used as desk toys

Introduction to the R package TDA

We present a short tutorial and introduction to using the R package TDA, which provides some tools for Topological Data Analysis. In particular, it includes implementations of functions that, given some data, provide topological information about the underlying space, such as the distance function, the distance to a measure, the kNN density estimator, the kernel density estimator, and the kernel distance. The salient topological features of the sublevel sets (or superlevel sets) of these functions can be quantified with persistent homology. We provide an R interface for the efficient algorithms of the C++ libraries GUDHI, Dionysus and PHAT, including a function for the persistent homology of the Rips filtration, and one for the persistent homology of sublevel sets (or superlevel sets) of arbitrary functions evaluated over a grid of points. The significance of the features in the resulting persistence diagrams can be analyzed with functions that implement recently developed statistical methods. The R package TDA also includes the implementation of an algorithm for density clustering, which allows us to identify the spatial organization of the probability mass associated to a density function and visualize it by means of a dendrogram, the cluster tree

On the Expressivity of Persistent Homology in Graph Learning

Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks

Vietoris-Rips complex의 최소 표본점과 persistence에 대하여

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2021.8. Otto van Koert.최근 위상적 자료분석 방법은 데이터 분석에 각광받고 있다. 이 논문에서는 데이터의 모형을 분석하기 위하여 Vietoris-Rips complex와 persistent 호몰로지를 연구하였다. 특별히, Vietoris-Rips complex구조가 주어진 데이터가 n차원 구와 같은 호몰로지를 갖을 수 있다고 할 때, 필요한 최소한의 데이터 양을 산출해냈다.Recently topological data analysis become a popular tool to analyze data. In this paper, we study the behaviour of Vietoris-Rips complex with persistent homology to figure out the shape of data. In particular, we find the minimal number of data points on a sphere such that homology of the Vietoris-Rips complex of those data points is isomorphic to the homology of the sphere.1 Introduction 1 2 Preliminaries 4 2.1 Nerve theorem with Vietoris-Rips complex 15 3 Persistence 17 3.1 Persistent homology 17 3.1.1 Tameness and barcodes 19 3.2 The Isometry theorem 25 4 2n-problem 29 4.1 Examples 30 4.2 Minimal Construction for S^2 33 4.3 Another proof of Minimal Construction for S^2 42 4.4 6 points probability for S^2 44 4.4.1 Script for 6 points probability 44 4.4.2 Bootstrap Con fidence Intervals 47 4.5 Vietoris-Rips complex for S^n 49 5 The Vietoris-Rips complex on a circle S^1 53 6 Reliable barcodes 59 6.1 On the length of barcodes 59 6.1.1 Mission impossible 60 6.1.2 Basic assumptions 61 6.1.3 Further assumptions 61 6.1.4 Convex balls and curvature 62 6.1.5 Background from metric geometry 65 6.1.6 Persistent homology of the Vietoris-Rips complex 67 6.2 Application to data 67 6.2.1 Revisiting the cube 68 6.2.2 Discretized curvature 69 7 Appendix 73 7.1 Notation and Conventions 73 7.2 Background from Probability 78 7.3 Scripts to compute persistent homology 82 Bibliography 84 Abstract (in Korean) 87박

On topological data analysis for structural dynamics: an introduction to persistent homology

Topological methods can provide a way of proposing new metrics and methods of scrutinising data, that otherwise may be overlooked. In this work, a method of quantifying the shape of data, via a topic called topological data analysis will be introduced. The main tool within topological data analysis (TDA) is persistent homology. Persistent homology is a method of quantifying the shape of data over a range of length scales. The required background and a method of computing persistent homology is briefly discussed in this work. Ideas from topological data analysis are then used for nonlinear dynamics to analyse some common attractors, by calculating their embedding dimension, and then to assess their general topologies. A method will also be proposed, that uses topological data analysis to determine the optimal delay for a time-delay embedding. TDA will also be applied to a Z24 Bridge case study in structural health monitoring, where it will be used to scrutinise different data partitions, classified by the conditions at which the data were collected. A metric, from topological data analysis, is used to compare data between the partitions. The results presented demonstrate that the presence of damage alters the manifold shape more significantly than the effects present from temperature

Distributing Persistent Homology via Spectral Sequences

We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of using them. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self contained presentation of the Mayer-Vietoris spectral sequence. Then we study the Persistent Mayer-Vietoris spectral sequence and present a solution to the extension problem. Finally, we review PerMaViss, a method implementing these ideas. This procedure distributes simplicial data, while focusing on merging homological information.Comment: Comments: 31 pages, 14 figures, 1 algorithm. Changes to previous version: longer introduction added, some material has been removed due to length constraints, section 5.2 added describing the procedure of computing the persistence Mayer-Vietoris spectral sequence, followed by complexity estimates in section 5.

Towards Emotion Recognition: A Persistent Entropy Application

Emotion recognition and classification is a very active area of research. In this paper, we present a first approach to emotion classification using persistent entropy and support vector machines. A topology-based model is applied to obtain a single real number from each raw signal. These data are used as input of a support vector machine to classify signals into 8 different emotions (calm, happy, sad, angry, fearful, disgust and surprised)