Topological Data Analysis - The Abel Symposium 2018

Series Foreword......Page 6
Preface......Page 7
References......Page 10
Contents......Page 13
A Fractal Dimension for Measures via Persistent Homology......Page 15
1 Introduction......Page 16
2.1 Minimal Spanning Trees......Page 17
2.4 Persistent Homology......Page 18
3.1 Fractal Dimensions......Page 20
3.2 Persistent Homology......Page 22
4 Definition of the Persistent Homology Fractal Dimension for Measures......Page 25
5 Experiments......Page 26
5.1 Estimates of Persistent Homology Fractal Dimensions......Page 27
5.2 Randomly Sampling from Self-Similar Fractals......Page 31
6 Limiting Distributions......Page 33
6.1 The Uniform Distribution on the Interval......Page 34
6.2 Experimental Evidence for Conjecture 2 in the Plane......Page 35
6.3 Examples where a Limiting Distribution Does Not Exist......Page 36
7 Another Way to Randomly Sample from the Sierpiński Triangle......Page 38
8 Asymptotic Approximation of the Scaling Exponent......Page 40
References......Page 42
1 Introduction......Page 46
2 Filtrations and Interleaving Distance......Page 49
3.1 Definition......Page 51
3.2 Stability......Page 54
3.3 Weighted Vietoris–Rips Filtrations......Page 55
4.1 The Distance to Measure (DTM)......Page 58
4.2 DTM-Filtrations......Page 59
4.3 Stability when p=1......Page 62
4.4 Stability when p>1......Page 66
4.5 Proof of Lemma 1 and Proposition 13......Page 71
5 Conclusion......Page 74
6 Supplementary Results for Sect.3......Page 75
7 Supplementary Results for Sect.4......Page 76
References......Page 79
1 Introduction......Page 80
1.1 Reformulation of the Induced Matching Theorem......Page 82
1.2 Reformulation of the Algebraic Stability Theorem......Page 83
1.3 Directly Constructing Barcodes as Matching Diagrams......Page 84
2.1 Properties of Mch and MchR......Page 85
2.2 Barcodes......Page 86
2.3 The Category of Barcodes......Page 87
2.4.1 A Functor from Barcodes to Diagrams......Page 88
2.4.2 A Functor from Diagrams to Barcodes......Page 89
2.5 Kernels, Cokernels, and Images of Barcodes......Page 90
3.1 Concrete Formulation of the Induced Matching Theorem......Page 92
3.2 Matchings Induced by Monos and Epis of Persistence Modules......Page 93
3.3 A Characterization of Morphisms with δ-Trivial (Co)kernel......Page 94
3.3.1 Shifts of R-Indexed Diagrams and Barcodes......Page 95
3.4 Proof of the Induced Matching Theorem......Page 97
4 Interleavings of Barcodes and the Bottleneck Distance......Page 98
4.1.3 Interleavings and Smallness of Kernels......Page 99
4.2.2 Interleaving Distance Equals Bottleneck Distance on Barcodes......Page 100
5 Constructing Barcodes and Induced Matchings in MchR......Page 102
References......Page 108
1 Introduction......Page 110
1.1 Related Work......Page 111
2.1 Persistence Modules, Persistence Diagrams, and Bar Codes......Page 112
2.2 Persistence Landscapes and Average Persistence Landscapes......Page 113
2.4.1 Invertibility......Page 114
2.4.3 The Persistence Landscape Kernel......Page 115
2.4.8 Confidence Bands for the Persistence Landscape......Page 116
3 Weighted Persistence Landscapes......Page 117
4 Persistence Landscapes as Tropical Rational Functions......Page 119
5 Reconstruction of Diagrams from an Average Persistence Landscape......Page 120
5.2 Bipartite Graph of a Persistence Diagram......Page 121
5.3 Critical Points of Persistence Landscapes......Page 122
5.5 Reconstruction of Persistence Diagrams from an Average Landscape......Page 123
5.7 Genericity of Arithmetically Independent Persistence Diagrams......Page 124
6 Metric Comparison of Persistence Landscapes and Persistence Diagrams......Page 126
References......Page 127
1 Introduction......Page 131
2 Neural Nets......Page 133
3 Natural Images and Convolutional Neural Nets......Page 140
4 Findings......Page 142
5.1 Generalities......Page 150
5.2 Data-analytically Defined Geometries......Page 152
5.3 Purely Data Driven Geometries......Page 155
References......Page 157
1 Introduction......Page 159
2.1 Background......Page 160
2.2 Preprocessing and Comparison Procedures......Page 161
3.1 Background......Page 163
3.2 Statistics on 1-dimensional Mappers......Page 164
3.3 Extension to Multivariate Mappers......Page 167
4.1 Method......Page 168
4.3 Formal Encoding and Statistical Significance......Page 170
References......Page 172
1 Introduction......Page 175
2 Neural Rings and the Pullback Map......Page 178
2.1 The Pullback Correspondence: A Closer Look......Page 179
3 Neural Ring Homomorphisms......Page 182
3.1 Proof of Theorem 3.4......Page 184
4 Neural Ring Homomorphisms and Convexity......Page 189
References......Page 192
1 Introduction......Page 193
2 Poisson–Delaunay Mosaics......Page 196
3 Poisson–Wrap Complexes......Page 202
4 Weighted Poisson–Delaunay Mosaics......Page 205
5 Weighted Poisson–Voronoi Tessellations......Page 212
6 Discussion......Page 216
Appendix 1: Voronoi Tessellations and Delaunay Mosaics......Page 218
Appendix 2: Discrete Morse Theory and Homology......Page 222
Appendix 3: Randomness and Expectation......Page 226
References......Page 228
Iterated Integrals and Population Time Series Analysis......Page 231
1 Path Signatures as Iterated Integrals......Page 232
2 Applications to Time Series Analysis......Page 239
2.1 Cyclicity and Lead-Lag Relationships......Page 241
2.2 Causality Analysis......Page 246
3 Generalizations and Outlook......Page 252
Appendix: Path Space Cochains......Page 254
References......Page 257
Prediction in Cancer Genomics Using Topological Signatures and Machine Learning......Page 259
1 Introduction......Page 260
2.1 Data......Page 261
2.1.3 Climent Data Set......Page 262
2.2.1 Foundations of Topological Data Analysis......Page 263
2.2.2 Topological Analysis of Array CGH (TAaCGH)......Page 265
2.2.3 Using Machine Learning for Patient Classification......Page 267
3.1 Computer Simulations of TAaCGH......Page 271
3.2 A Logistic Model for ERBB2+ Breast Cancer......Page 274
3.3 A Logistic Regression Model for ER+ Breast Cancer......Page 275
4 Discussion......Page 277
Appendix......Page 280
References......Page 286
1 Introduction......Page 289
2.1 The Blue Brain Project......Page 291
2.3 Structural Insights......Page 292
2.4 Functional Insights......Page 297
2.5 Perspectives......Page 299
3 An Objective Topological Descriptor of Neuron Morphologies......Page 301
3.1 The Topological Morphology Descriptor......Page 303
3.2 Properties of the TMD......Page 305
3.3 Objective Classification of Rat Pyramidal Cells......Page 309
3.3.1 Methods......Page 311
3.3.2 Results......Page 314
3.4 Perspectives......Page 315
References......Page 316
1.1 Background......Page 318
1.2 Main Results......Page 320
2.1 Preliminaries......Page 321
2.2 Hole Graph......Page 325
2.3 Face Percolation......Page 326
2.4 Main Theorems......Page 329
3.1 Bond Percolation on the Dual Lattice......Page 330
3.2 Estimates of the Critical Probability......Page 336
4 Uniqueness of the Infinite Hole Cluster......Page 338
5.1 The Right Continuity of θhole(p)......Page 343
5.2 Left Continuity of θhole(p) in the Supercritical Phase......Page 346
5.3 The Number of Vertices in the Hole Graph......Page 348
5.4 The Size of Holes......Page 349
6 Conclusions......Page 351
References......Page 353
1 Introduction......Page 354
2.1 Stable Hyperplane Codes......Page 355
2.2 Bitflips and Stable Hyperplane Codes......Page 357
2.3 The Polar Complex......Page 358
3.1 Local Obstructions and Bitflips......Page 360
3.2 Spherical Link Obstructions......Page 362
3.3 Chamber Obstructions......Page 363
4 The Main Results......Page 365
5 Discussion......Page 366
6.1 The Neural and the Stanley-Reisner Ideal......Page 367
6.2 Sphere Link Obstructions and Multigraded Free Resolutions......Page 370
7.1 Shellability......Page 372
7.2 Obstructions Following from Shellability......Page 376
References......Page 379
1 Introduction......Page 381
1.1 Overview of the Results in [26]......Page 382
2.1 Examples of DMSs and Tameness......Page 384
2.2 The λ-Slack Interleaving Distance Between DMSs......Page 385
3 Dynamic Graphs (DGs)......Page 387
3.1 Definition of DGs......Page 388
3.2 The Interleaving Distance Between DGs......Page 389
3.3 From DMSs to DGs......Page 391
4.1.1 Formigrams......Page 392
4.3 From DGs to Formigrams......Page 394
5.1 Boids Model: Creating Synthetic Flocking Behaviors......Page 395
5.3 Classification of Flocking Behaviors......Page 396
References......Page 398
1 Introduction......Page 400
2 Bisheaves Around Simplicial Complexes......Page 402
2.2 Bisheaves from Fibers......Page 403
3 Stratifications Along Bisheaves......Page 404
4.1 Localizations of the Face Poset......Page 406
4.2 Top Strata......Page 407
4.3 Lower Strata......Page 409
References......Page 412
1 Introduction......Page 413
2 Background......Page 415
3.1 Persistent Moore Spaces......Page 419
3.2 Point Cloud Continuation......Page 424
3.3 Functional Optimization and Continuation......Page 426
4 Persistence and Left Inverses......Page 428
4.1 Extrinsic Persistent Homology Transforms......Page 429
4.1.1 How Many Directions Suffice?......Page 432
4.2 Intrinsic Persistent Homology Transform......Page 434
References......Page 439
1 Introduction......Page 442
1.1 Our Contribution......Page 443
1.2 The Sparse Circular Coordinates Algorithm......Page 444
2.1 Principal Bundles......Page 445
3 From Integer Simplicial Cohomology to Circular Coordinates......Page 449
4.1 Landmark Selection......Page 452
4.4 Lifting Persistence to Integer Coefficients......Page 453
4.5 Use Rips, Not Nerves......Page 454
4.6 Harmonic Smoothing......Page 455
5.1.1 A Noisy Circle......Page 458
5.1.3 The Klein Bottle......Page 459
5.2.1 COIL 20......Page 460
5.2.2 The Mumford Data......Page 461
6 Discussion......Page 463
References......Page 464
1 Introduction......Page 466
2.1 Topological Summary Statistics......Page 468
2.2 Distance Correlation......Page 470
2.3 Metric Spaces of Strong Negative Type......Page 474
3 A Veritable Zoo of Topological Summaries, Some of Which Are of Strong Negative Type......Page 476
3.2 Persistence Diagrams......Page 477
3.3 Persistence Landscapes......Page 483
3.4 Persistence Scale Space Kernel......Page 485
3.5 Sliced Wasserstein Kernel Distance......Page 486
4.1 Erdős–Rényi......Page 487
4.3 Geometric Random Complexes for Points Sampled on a Torus......Page 488
4.4 Geometric Random Complexes for Points Sampled from a Unit Cube......Page 490
5 Distance Correlation to Another Parameter......Page 491
5.1 Parameterised Interpolation Between Erdős–Rényi and Geometric Complexes......Page 492
5.2 Digital Elevation Models and Terrain Ruggedness......Page 493
6 Future Directions......Page 495
References......Page 496
1 Introduction......Page 498
1.2 Structure of This Paper......Page 501
2 Nerve Lemmata and Topological Background......Page 502
3 Obstructions and Certified Mapper......Page 504
4 Statistical Acyclicity......Page 506
4.1 Null Models......Page 507
4.2.1 Normal Approximation of Maximal Ratios......Page 508
4.2.2 Normal Approximation of Quantiles......Page 509
4.3 Empirical Distributions and Normalized Maximal Persistences......Page 510
5 Experiments......Page 511
6.1 Validation......Page 512
6.2 Power Estimation......Page 513
6.4 Real World Data......Page 515
7.1 Multiple Testing Paradigms......Page 518
7.2 Certified Mapper......Page 519
8 Future Directions......Page 520
References......Page 521