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
