Computational topology for data analysis:
"In this chapter, we introduce some of the very basics that are used throughout the book. First, we give the definition of a topological space and related notions of open and closed sets, covers, subspace topology. To connect topology and geometry, we devote a section on metric spaces. Maps suc...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Cambridge University Press
2022
|
| Ausgabe: | First published |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "In this chapter, we introduce some of the very basics that are used throughout the book. First, we give the definition of a topological space and related notions of open and closed sets, covers, subspace topology. To connect topology and geometry, we devote a section on metric spaces. Maps such as homeomorphism and homotopy equivalence that play a significant role to relate topological spaces. Certain categories of topological spaces become important for their wide presence in applications. Manifolds are one such category which we introduce in this chapter. Functions on them satisfying certain conditions are presented as Morse functions. The critical points of such functions relate to the topology of the manifold they are defined on. We introduce these concepts in the smooth setting in this chapter, and later adapt them for the piecewise linear domains frequently used for finite computations. Finally, a section on Notes points out to the history and relevant literature for the concepts delineated in the chapter. It ends with a series of exercises that may be used for teaching a class on the subject both at graduate and undergraduate level |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xix, 433 Seiten Illustrationen, Diagramme |
Internformat
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| adam_text | Contents page xi Preface Prelude XV Basics 1.1 Topological Space 1.2 Metric Space Topology 1.3 Maps, Homeomorphisms, and Homotopies 1.4 Manifolds 1.4.1 Smooth Manifolds 1.5 Functions on Smooth Manifolds 1.5.1 Gradients and Critical Points 1.5.2 Morse Functions and Morse Lemma 1.5.3 Connection to Topology 1.6 Notes and Exercises 1 1 5 9 14 16 17 17 20 22 24 Complexes and Homology Groups 2.1 Simplicial Complex 2.2 Nerves, Cech and Rips Complexes 2.3 Sparse Complexes 2.3.1 Delaunay Complex 2.3.2 Witness Complex 2.3.3 Graph Induced Complex 2.4 Chains, Cycles, Boundaries 2.4.1 Algebraic Structures 2.4.2 Chains 2.4.3 Boundaries and Cycles 2.5 Homology 2.5.1 Induced Homology 2.5.2 Relative Homology 26 27 31 33 34 36 38 43 43 45 46 49 51 52 V
Contents VI 2.6 2.5.3 Singular Homology 2.5.4 Cohomology Notes and Exercises 54 55 57 3 Topological Persistence 3.1 Filtrations and Persistence 3.1.1 Space Filtration 3.1.2 Simplicial Filtrations and Persistence 3.2 Persistence 3.2.1 Persistence Diagram 3.3 Persistence Algorithm 3.3.1 Matrix Reduction Algorithm 3.3.2 Efficient Implementation 3.4 Persistence Modules 3.5 Persistence for PL-Functions 3.5.1 PL-Functions and Critical Points 3.5.2 Lower-Star Filtration and Its Persistent Homology 3.5.3 Persistence Algorithm for Zeroth Persistent Homology 3.6 Notes and Exercises 60 62 62 63 68 70 76 80 86 90 95 96 101 104 108 4 General Persistence 4.1 Stability of Towers 4.2 Computing Persistence of Simplicial Towers 4.2.1 Annotations 4.2.2 Algorithm 4.2.3 Elementary Inclusion 4.2.4 Elementary Collapse 4.3 Persistence for Zigzag Filtration 4.3.1 Approach 4.3.2 Zigzag Persistence Algorithm 4.4 Persistence for Zigzag Towers 4.5 Levelset Zigzag Persistence 4.5.1 Simplicial Levelset Zigzag Filtration 4.5.2 Barcode for Levelset Zigzag Filtration 4.5.3 Correspondence to Sublevel Set Persistence 4.5.4 Correspondence to Extended Persistence 4.6 Notes and Exercises 112 113 117 117 118 119 121 125 128 131 134 137 139 141 143 143 145 5 Generators and Optimality 5.1 Optimal Generators/Basis 5.1.1 Greedy Algorithm for Optimal Hp(ÄT)-Basis 148 150 151
Contents 5.2 5.3 5.4 5.1.2 Optimal Hi (Äľ)-Basis and Independence Check Localization 5.2.1 Linear Program 5.2.2 Total Unimodularity 5.2.3 Relative Torsion Persistent Cycles 5.3.1 Finite Intervals for Weak (p + l)-Pseudomanifolds 5.3.2 Algorithm Correctness 5.3.3 Infinite Intervals for Weak (p + 1)Pseudomanifolds Embedded in RP+1 Notes and Exercises vii 155 157 159 161 162 165 167 172 174 175 6 Topological Analysis of Point Clouds 6.1 Persistence for Rips and Cech Filtrations 6.2 Approximation via Data Sparsification 6.2.1 Data Sparsification for Rips Filtration via Reweighting 6.2.2 Approximation via Simplicial Tower 6.3 Homology Inference from Point Cloud Data 6.3.1 Distance Field and Feature Sizes 6.3.2 Data on a Manifold 6.3.3 Data on a Compact Set 6.4 Homology Inference for Scalar Fields 6.4.1 Problem Setup 6.4.2 Inference Guarantees 6.5 Notes and Exercises 178 179 182 183 190 192 193 195 197 198 199 201 204 7 Reeb Graphs 7.1 Reeb Graph: Definitions and Properties 7.2 Algorithms in the PL-Setting 7.2.1 An 0(m logm)-Time Algorithm via Dynamic Graph Connectivity 7.2.2 A Randomized Algorithm with 0{m log ni) Expected Time 7.2.3 Homology Groups of Reeb Graphs 7.3 Distances for Reeb Graphs 7.3.1 Interleaving Distance 7.3.2 Functional Distortion Distance 7.4 Notes and Exercises 207 208 211 213 217 220 223 224 226 229
viii Contents 8 Topological Analysis of Graphs 8.1 Topological Summaries for Graphs 8.1.1 Combinatorial Graphs 8.1.2 Graphs Viewed as Metric Spaces 8.2 Graph Comparison 8.3 Topological Invariants for Directed Graphs 8.3.1 Simplicial Complexes for Directed Graphs 8.3.2 Path Homology for Directed Graphs 8.3.3 Computation of (Persistent) Path Homology 8.4 Notes and Exercises 233 234 235 236 239 240 241 243 245 252 9 Cover, Nerve, and Mapper 9.1 Covers and Nerves 9.1.1 Special Case of Hi 9.2 Analysis of Persistent Hi-Classes 9.3 Mapper and Multiscale Mapper 9.3.1 Multiscale Mapper 9.3.2 Persistence of Ні-Classes in Mapper and Multiscale Mapper 9.4 Stability 9.4.1 Interleaving of Cover Towers and Multiscale Mappers 9.4.2 (c, r)-Good Covers 9.4.3 Relation to Intrinsic Čech Filtration 9.5 Exact Computation for PL-Functions on Simplicial Domains 9.6 Approximating Multiscale Mapper for General Maps 9.6.1 Combinatorial Mapper and Multiscale Mapper 9.6.2 Advantage of Combinatorial Multiscale Mapper 9.7 Notes and Exercises 255 257 261 265 268 271 273 274 275 276 279 281 283 284 285 286 Discrete Morse Theory and Applications 10.1 Discrete Morse Function 10.1.1 Discrete Morse Vector Field 10.2 Persistence-Based Discrete Morse Vector Fields 10.2.1 Persistence-Guided Cancellation 10.2.2 Algorithms 10.3 Stable and Unstable Manifolds 10.3.1 Morse Theory Revisited 10.3.2 (Un)Stable Manifolds in Discrete Morse Vector Fields 10.4 Graph Reconstruction 10.4.1 Algorithm 10.4.2 Noise Model 289 290 292 295 295 298 303 303 304 305 306 308 10
Contents 11 12 ix 10.4.3 Theoretical Guarantees 10.5 Applications 10.5.1 Road Network 10.5.2 Neuron Network 10.6 Notes and Exercises 309 313 313 315 316 Multiparameter Persistence and Decomposition 11.1 Multiparameter Persistence Modules 11.1.1 Persistence Modules as Graded Modules 11.2 Presentations of Persistence Modules 11.2.1 Presentation and Its Decomposition 11.3 Presentation Matrix: Diagonalization and Simplification 11.3.1 Simplification 11.4 Total Diagonalization Algorithm 11.4.1 Running TotDiagonalize on the Working Example in Figure 11.5 11.5 Computing Presentations 11.5.1 Graded Chain, Cycle, and Boundary Modules 11.5.2 Multiparameter Filtration, Zero-Dimensional Homology 11.5.3 Two-Parameter Filtration, Multi-Dimensional Homology 11.5.4 ¿-Parameter (d 2) Filtration, Multi-Dimensional Homology 11.5.5 Time Complexity 11.6 Invariants 11.6.1 Rank Invariants 11.6.2 Graded Betti Numbers and Blockcodes 11.7 Notes and Exercises 321 325 325 328 329 332 333 337 Multiparameter Persistence and Distances 12.1 Persistence Modules from Categorical Viewpoint 12.2 Interleaving Distance 12.3 Matching Distance 12.3.1 Computing Matching Distance 12.4 Bottleneck Distance 12.4.1 Interval Decomposable Modules 12.4.2 Bottleneck Distance for Two-Parameter Interval Decomposable Modules 12.4.3 Algorithm to Compute d/ for Intervals 12.5 Notes and Exercises 365 367 369 370 371 374 376 347 350 350 353 353 354 356 356 357 358 361 378 383 386
x Contents 13 Topological Persistence and Machine Learning 13.1 Feature Vectorization of Persistence Diagrams 13.1.1 Persistence Landscape 13.1.2 Persistence Scale Space Kernel (PSSK) 13.1.3 Persistence Images 13.1.4 Persistence Weighted Gaussian Kernel(PWGK) 13.1.5 Sliced Wasserstein Kernel 13.1.6 Persistence Fisher Kernel 13.2 Optimizing Topological Loss Functions 13.2.1 Topological Regularizer 13.2.2 Gradients of a Persistence-Based Topological Function 13.3 Statistical Treatment of Topological Summaries 13.4 Bibliographical Notes 389 390 391 393 394 396 398 399 400 401 403 405 408 References Index 411 429
|
| adam_txt |
Contents page xi Preface Prelude XV Basics 1.1 Topological Space 1.2 Metric Space Topology 1.3 Maps, Homeomorphisms, and Homotopies 1.4 Manifolds 1.4.1 Smooth Manifolds 1.5 Functions on Smooth Manifolds 1.5.1 Gradients and Critical Points 1.5.2 Morse Functions and Morse Lemma 1.5.3 Connection to Topology 1.6 Notes and Exercises 1 1 5 9 14 16 17 17 20 22 24 Complexes and Homology Groups 2.1 Simplicial Complex 2.2 Nerves, Cech and Rips Complexes 2.3 Sparse Complexes 2.3.1 Delaunay Complex 2.3.2 Witness Complex 2.3.3 Graph Induced Complex 2.4 Chains, Cycles, Boundaries 2.4.1 Algebraic Structures 2.4.2 Chains 2.4.3 Boundaries and Cycles 2.5 Homology 2.5.1 Induced Homology 2.5.2 Relative Homology 26 27 31 33 34 36 38 43 43 45 46 49 51 52 V
Contents VI 2.6 2.5.3 Singular Homology 2.5.4 Cohomology Notes and Exercises 54 55 57 3 Topological Persistence 3.1 Filtrations and Persistence 3.1.1 Space Filtration 3.1.2 Simplicial Filtrations and Persistence 3.2 Persistence 3.2.1 Persistence Diagram 3.3 Persistence Algorithm 3.3.1 Matrix Reduction Algorithm 3.3.2 Efficient Implementation 3.4 Persistence Modules 3.5 Persistence for PL-Functions 3.5.1 PL-Functions and Critical Points 3.5.2 Lower-Star Filtration and Its Persistent Homology 3.5.3 Persistence Algorithm for Zeroth Persistent Homology 3.6 Notes and Exercises 60 62 62 63 68 70 76 80 86 90 95 96 101 104 108 4 General Persistence 4.1 Stability of Towers 4.2 Computing Persistence of Simplicial Towers 4.2.1 Annotations 4.2.2 Algorithm 4.2.3 Elementary Inclusion 4.2.4 Elementary Collapse 4.3 Persistence for Zigzag Filtration 4.3.1 Approach 4.3.2 Zigzag Persistence Algorithm 4.4 Persistence for Zigzag Towers 4.5 Levelset Zigzag Persistence 4.5.1 Simplicial Levelset Zigzag Filtration 4.5.2 Barcode for Levelset Zigzag Filtration 4.5.3 Correspondence to Sublevel Set Persistence 4.5.4 Correspondence to Extended Persistence 4.6 Notes and Exercises 112 113 117 117 118 119 121 125 128 131 134 137 139 141 143 143 145 5 Generators and Optimality 5.1 Optimal Generators/Basis 5.1.1 Greedy Algorithm for Optimal Hp(ÄT)-Basis 148 150 151
Contents 5.2 5.3 5.4 5.1.2 Optimal Hi (Äľ)-Basis and Independence Check Localization 5.2.1 Linear Program 5.2.2 Total Unimodularity 5.2.3 Relative Torsion Persistent Cycles 5.3.1 Finite Intervals for Weak (p + l)-Pseudomanifolds 5.3.2 Algorithm Correctness 5.3.3 Infinite Intervals for Weak (p + 1)Pseudomanifolds Embedded in RP+1 Notes and Exercises vii 155 157 159 161 162 165 167 172 174 175 6 Topological Analysis of Point Clouds 6.1 Persistence for Rips and Cech Filtrations 6.2 Approximation via Data Sparsification 6.2.1 Data Sparsification for Rips Filtration via Reweighting 6.2.2 Approximation via Simplicial Tower 6.3 Homology Inference from Point Cloud Data 6.3.1 Distance Field and Feature Sizes 6.3.2 Data on a Manifold 6.3.3 Data on a Compact Set 6.4 Homology Inference for Scalar Fields 6.4.1 Problem Setup 6.4.2 Inference Guarantees 6.5 Notes and Exercises 178 179 182 183 190 192 193 195 197 198 199 201 204 7 Reeb Graphs 7.1 Reeb Graph: Definitions and Properties 7.2 Algorithms in the PL-Setting 7.2.1 An 0(m logm)-Time Algorithm via Dynamic Graph Connectivity 7.2.2 A Randomized Algorithm with 0{m log ni) Expected Time 7.2.3 Homology Groups of Reeb Graphs 7.3 Distances for Reeb Graphs 7.3.1 Interleaving Distance 7.3.2 Functional Distortion Distance 7.4 Notes and Exercises 207 208 211 213 217 220 223 224 226 229
viii Contents 8 Topological Analysis of Graphs 8.1 Topological Summaries for Graphs 8.1.1 Combinatorial Graphs 8.1.2 Graphs Viewed as Metric Spaces 8.2 Graph Comparison 8.3 Topological Invariants for Directed Graphs 8.3.1 Simplicial Complexes for Directed Graphs 8.3.2 Path Homology for Directed Graphs 8.3.3 Computation of (Persistent) Path Homology 8.4 Notes and Exercises 233 234 235 236 239 240 241 243 245 252 9 Cover, Nerve, and Mapper 9.1 Covers and Nerves 9.1.1 Special Case of Hi 9.2 Analysis of Persistent Hi-Classes 9.3 Mapper and Multiscale Mapper 9.3.1 Multiscale Mapper 9.3.2 Persistence of Ні-Classes in Mapper and Multiscale Mapper 9.4 Stability 9.4.1 Interleaving of Cover Towers and Multiscale Mappers 9.4.2 (c, r)-Good Covers 9.4.3 Relation to Intrinsic Čech Filtration 9.5 Exact Computation for PL-Functions on Simplicial Domains 9.6 Approximating Multiscale Mapper for General Maps 9.6.1 Combinatorial Mapper and Multiscale Mapper 9.6.2 Advantage of Combinatorial Multiscale Mapper 9.7 Notes and Exercises 255 257 261 265 268 271 273 274 275 276 279 281 283 284 285 286 Discrete Morse Theory and Applications 10.1 Discrete Morse Function 10.1.1 Discrete Morse Vector Field 10.2 Persistence-Based Discrete Morse Vector Fields 10.2.1 Persistence-Guided Cancellation 10.2.2 Algorithms 10.3 Stable and Unstable Manifolds 10.3.1 Morse Theory Revisited 10.3.2 (Un)Stable Manifolds in Discrete Morse Vector Fields 10.4 Graph Reconstruction 10.4.1 Algorithm 10.4.2 Noise Model 289 290 292 295 295 298 303 303 304 305 306 308 10
Contents 11 12 ix 10.4.3 Theoretical Guarantees 10.5 Applications 10.5.1 Road Network 10.5.2 Neuron Network 10.6 Notes and Exercises 309 313 313 315 316 Multiparameter Persistence and Decomposition 11.1 Multiparameter Persistence Modules 11.1.1 Persistence Modules as Graded Modules 11.2 Presentations of Persistence Modules 11.2.1 Presentation and Its Decomposition 11.3 Presentation Matrix: Diagonalization and Simplification 11.3.1 Simplification 11.4 Total Diagonalization Algorithm 11.4.1 Running TotDiagonalize on the Working Example in Figure 11.5 11.5 Computing Presentations 11.5.1 Graded Chain, Cycle, and Boundary Modules 11.5.2 Multiparameter Filtration, Zero-Dimensional Homology 11.5.3 Two-Parameter Filtration, Multi-Dimensional Homology 11.5.4 ¿-Parameter (d 2) Filtration, Multi-Dimensional Homology 11.5.5 Time Complexity 11.6 Invariants 11.6.1 Rank Invariants 11.6.2 Graded Betti Numbers and Blockcodes 11.7 Notes and Exercises 321 325 325 328 329 332 333 337 Multiparameter Persistence and Distances 12.1 Persistence Modules from Categorical Viewpoint 12.2 Interleaving Distance 12.3 Matching Distance 12.3.1 Computing Matching Distance 12.4 Bottleneck Distance 12.4.1 Interval Decomposable Modules 12.4.2 Bottleneck Distance for Two-Parameter Interval Decomposable Modules 12.4.3 Algorithm to Compute d/ for Intervals 12.5 Notes and Exercises 365 367 369 370 371 374 376 347 350 350 353 353 354 356 356 357 358 361 378 383 386
x Contents 13 Topological Persistence and Machine Learning 13.1 Feature Vectorization of Persistence Diagrams 13.1.1 Persistence Landscape 13.1.2 Persistence Scale Space Kernel (PSSK) 13.1.3 Persistence Images 13.1.4 Persistence Weighted Gaussian Kernel(PWGK) 13.1.5 Sliced Wasserstein Kernel 13.1.6 Persistence Fisher Kernel 13.2 Optimizing Topological Loss Functions 13.2.1 Topological Regularizer 13.2.2 Gradients of a Persistence-Based Topological Function 13.3 Statistical Treatment of Topological Summaries 13.4 Bibliographical Notes 389 390 391 393 394 396 398 399 400 401 403 405 408 References Index 411 429 |
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| owner | DE-739 DE-521 |
| owner_facet | DE-739 DE-521 |
| physical | xix, 433 Seiten Illustrationen, Diagramme |
| publishDate | 2022 |
| publishDateSearch | 2022 |
| publishDateSort | 2022 |
| publisher | Cambridge University Press |
| record_format | marc |
| spelling | Dey, Tamal 1964- Verfasser (DE-588)138060134 aut Computational topology for data analysis Tamal Krishna Dey, Department of Computer Science, Purdue University, Yusu Wang, University of California, San Diego First published New York Cambridge University Press 2022 xix, 433 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index "In this chapter, we introduce some of the very basics that are used throughout the book. First, we give the definition of a topological space and related notions of open and closed sets, covers, subspace topology. To connect topology and geometry, we devote a section on metric spaces. Maps such as homeomorphism and homotopy equivalence that play a significant role to relate topological spaces. Certain categories of topological spaces become important for their wide presence in applications. Manifolds are one such category which we introduce in this chapter. Functions on them satisfying certain conditions are presented as Morse functions. The critical points of such functions relate to the topology of the manifold they are defined on. We introduce these concepts in the smooth setting in this chapter, and later adapt them for the piecewise linear domains frequently used for finite computations. Finally, a section on Notes points out to the history and relevant literature for the concepts delineated in the chapter. It ends with a series of exercises that may be used for teaching a class on the subject both at graduate and undergraduate level Topology Topologie (DE-588)4060425-1 gnd rswk-swf Datenanalyse (DE-588)4123037-1 gnd rswk-swf Topologie (DE-588)4060425-1 s Datenanalyse (DE-588)4123037-1 s DE-604 Wang, Yusu 1976- Sonstige (DE-588)1253963703 oth Erscheint auch als Online-Ausgabe 978-1-00-909995-0 (DE-604)BV047926559 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033232792&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dey, Tamal 1964- Computational topology for data analysis Topology Topologie (DE-588)4060425-1 gnd Datenanalyse (DE-588)4123037-1 gnd |
| subject_GND | (DE-588)4060425-1 (DE-588)4123037-1 |
| title | Computational topology for data analysis |
| title_auth | Computational topology for data analysis |
| title_exact_search | Computational topology for data analysis |
| title_exact_search_txtP | Computational topology for data analysis |
| title_full | Computational topology for data analysis Tamal Krishna Dey, Department of Computer Science, Purdue University, Yusu Wang, University of California, San Diego |
| title_fullStr | Computational topology for data analysis Tamal Krishna Dey, Department of Computer Science, Purdue University, Yusu Wang, University of California, San Diego |
| title_full_unstemmed | Computational topology for data analysis Tamal Krishna Dey, Department of Computer Science, Purdue University, Yusu Wang, University of California, San Diego |
| title_short | Computational topology for data analysis |
| title_sort | computational topology for data analysis |
| topic | Topology Topologie (DE-588)4060425-1 gnd Datenanalyse (DE-588)4123037-1 gnd |
| topic_facet | Topology Topologie Datenanalyse |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033232792&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT deytamal computationaltopologyfordataanalysis AT wangyusu computationaltopologyfordataanalysis |