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E-raamat: Persistence Theory: From Quiver Representations to Data Analysis

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Persistence Theory: From Quiver Representations to Data AnalysisSuurem pilt
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Persistence theory emerged in the early 2000s as a new theory in the area of applied and computational topology. This book provides a broad and modern view of persistence theory, including its algebraic, topological, and algorithmic aspects. It also elaborates on applications in data analysis. The level of detail of the exposition has been set so as to keep a survey style, while providing sufficient insights into the proofs so the reader can understand the mechanisms at work.

The book is organized into three parts. The first part is dedicated to the foundations of persistence and emphasizes its connection to quiver representation theory. The second part focuses on its connection to applications through a few selected topics. The third part provides perspectives for both the theory and its applications. It can be used as a text for a course on applied topology, on data analysis, or on applied statistics.
Preface vii
Introduction 1(10)
Part 1 Theoretical Foundations
11(54)
Chapter 1 Algebraic Persistence
13(16)
1 A quick walk through the theory of quiver representations
14(5)
2 Persistence modules and interval decompositions
19(2)
3 Persistence barcodes and diagrams
21(4)
4 Extension to interval-indecomposable persistence modules
25(1)
5 Discussion
26(3)
Chapter 2 Topological Persistence
29(20)
1 Topological constructions
29(10)
2 Calculations
39(10)
Chapter 3 Stability
49(16)
1 Metrics
50(4)
2 Proof of the stability part of the Isometry Theorem
54(6)
3 Proof of the converse stability part of the Isometry Theorem
60(1)
4 Discussion
61(4)
Part 2 Applications
65(88)
Chapter 4 Topological Inference
67(18)
1 Inference using distance functions
71(7)
2 From offsets to filtrations
78(3)
3 From filtrations to simplicial filtrations
81(4)
Chapter 5 Topological Inference 2.0
85(30)
1 Simple geometric predicates
88(4)
2 Linear size
92(3)
3 Scaling up with the intrinsic dimensionality of the data
95(9)
4 Side-by-side comparison
104(2)
5 Natural images
106(4)
6 Dealing with outliers
110(5)
Chapter 6 Clustering
115(18)
1 Contributions of persistence
117(1)
2 ToMATo
118(4)
3 Theoretical guarantees
122(4)
4 Experimental results
126(3)
5 Higher-dimensional structure
129(4)
Chapter 7 Signatures for Metric Spaces
133(20)
1 Simplicial filtrations for arbitrary metric spaces
138(2)
2 Stability for finite metric spaces
140(2)
3 Stability for totally bounded metric spaces
142(4)
4 Signatures for metric spaces equipped with functions
146(1)
5 Computations
146(7)
Part 3 Perspectives
153(14)
Chapter 8 New Trends in Topological Data Analysis
155(8)
1 Optimized inference pipeline
156(2)
2 Statistical topological data analysis
158(2)
3 Topological data analysis and machine learning
160(3)
Chapter 9 Further prospects on the theory
163(4)
1 Persistence for other types of quivers
163(2)
2 Stability for zigzags
165(1)
3 Simplification and reconstruction
165(2)
Appendix A Introduction to Quiver Theory with a View Toward Persistence 167(1)
1 Quivers 168(1)
2 The category of quiver representations 168(2)
3 Classification of quiver representations 170(5)
4 Reflections 175(11)
5 Proof of Gabriel's theorem: the general case 186(5)
6 Beyond Gabriel's theorem 191(6)
Bibliography 197(16)
List of Figures 213(4)
Index 217
Steve Y. Oudot, Inria Saclay, Palaiseau, France.
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