Handbook of Variational Methods for Nonlinear Geometric Data

Preface......Page 5
Acknowledgements......Page 7
Introduction......Page 8
Contents......Page 15
Contributors......Page 18
About the Editors......Page 22
Part I Processing Geometric Data......Page 24
Contents......Page 25
1.1 Introduction......Page 26
1.2 Constructions of geometric Finite Elements......Page 27
1.2.1 Projection-Based Finite Elements......Page 28
1.2.1.1 Construction......Page 29
1.2.1.2 Properties......Page 30
1.2.2.1 Construction......Page 32
1.2.2.2 Properties......Page 34
1.2.2.3 Relationship to Projection-Based Finite Elements......Page 36
1.2.3.1 Construction......Page 37
1.2.4 Interpolation in Normal Coordinates......Page 39
1.2.4.1 Construction......Page 40
1.2.4.2 Properties......Page 41
1.3 Discrete Test Functions and Vector Field Interpolation......Page 42
1.3.1 Algebraic Representation of Test Functions......Page 43
1.3.2 Test Vector Fields as Discretizations of Maps into the Tangent Bundle......Page 44
1.4.1 Sobolev Spaces of Maps into Manifolds......Page 47
1.4.1.1 Smoothness Descriptors......Page 49
1.4.1.2 Scaling Properties......Page 50
1.4.1.3 Sobolev Distances......Page 51
1.4.2 Discretization of Elliptic Energy Minimization Problems......Page 53
1.4.2.1 Ellipticity......Page 54
1.4.2.2 Approximability Conditions......Page 55
1.4.2.3 W1,2-Error Bounds......Page 57
1.4.2.4 L2-Error Bounds......Page 58
1.4.3.1 Geodesic Finite Elements......Page 61
1.4.3.2 Projection-Based Finite Elements......Page 62
1.5.1 Harmonic Maps into the Sphere......Page 63
1.5.2 Magnetic Skyrmions in the Plane......Page 66
1.5.3 Geometrically Exact Cosserat Plates......Page 67
References......Page 69
Contents......Page 72
2.1 Introduction......Page 73
2.2 Total Variation Regularization of Manifold Valued Data......Page 74
2.2.1 Models......Page 75
2.2.2 Algorithmic Realization......Page 76
CPPAs and PPPAs......Page 77
Proximal Mappings for the Atoms of the TV Functionals......Page 79
2.3 Higher Order Total Variation Approaches, Total Generalized Variation......Page 82
2.3.1 Models......Page 84
Consistency......Page 88
Higher-Order Regularized Denoising......Page 90
2.3.2 Algorithmic Realization......Page 91
Numerical Examples......Page 93
2.4 Mumford-Shah Regularization for Manifold Valued Data......Page 94
2.4.1 Models......Page 95
Dynamic Programming Scheme......Page 97
Algorithms for the Univariate Mumford-Shah and Potts Problem......Page 98
Multivariate Mumford-Shah and Potts Problems......Page 99
2.5.1 Models......Page 101
Basic Generalized Forward Backward Scheme......Page 103
A Generalized Forward Backward Scheme with Gauß-Seidel Update and a Trajectory Method......Page 104
2.6 Wavelet Sparse Regularization of Manifold Valued Data......Page 106
2.6.1 Model......Page 107
2.6.2 Algorithmic Realization......Page 109
References......Page 110
Contents......Page 115
3.1 Introduction......Page 116
3.1.1 Functional Lifting in Euclidean Spaces......Page 117
3.1.2 Manifold-Valued Functional Lifting......Page 120
3.1.3 Further Related Work......Page 122
3.2.1 Calculus of Variations on Submanifolds......Page 123
3.2.2 Finite Elements on Submanifolds......Page 125
3.2.3 Relation to lellmann2013vogt......Page 127
3.2.4 Full Discretization and Numerical Implementation......Page 130
3.3.2 Three-Dimensional Manifolds: SO(3)......Page 131
3.3.3 Normals Fields from Digital Elevation Data......Page 132
3.3.4 Denoising of High Resolution InSAR Data......Page 134
References......Page 136
Contents......Page 140
4.1.1 The Fréchet Mean......Page 141
The Fréchet Mean in the Non-unique Case......Page 142
Properties of the Riemannian Exponential Mapping......Page 143
The Exponential Mapping in Symmetric Spaces......Page 144
Affine Averages with Respect to a Base Point......Page 145
4.2.1 Defining Stationary Subdivision......Page 146
Linear Subdivision Rules and Their Nonlinear Analogues......Page 147
Definition of Convergence......Page 149
4.2.3 Probabilistic Interpretation of Subdivision in Metric Spaces......Page 153
4.2.4 The Convergence Problem in Manifolds......Page 155
Subdivision in Riemannian Manifolds with Positive Curvature......Page 156
4.3 Smoothness Analysis of Subdivision Rules......Page 157
4.3.1 Derivatives of Limits......Page 158
Smoothness from Proximity......Page 159
Smoothness Equivalence......Page 160
4.3.3 Subdivision of Hermite Data......Page 161
4.3.4 Subdivision with Irregular Combinatorics......Page 162
4.4.1 Definition of Intrinsic Multiscale Transforms......Page 163
Characterizing Smoothness by Coefficient Decay......Page 166
Stability......Page 167
References......Page 168
5.1 Introduction......Page 172
5.2 Shape Space of Lipschitz Immersions......Page 178
5.3 Notions of Convergence for Variational Problems......Page 180
5.4 Practitioner's Guide to Kuratowski Convergence of Minimizers......Page 182
5.5 Convergence of Discrete Minimal Surfaces and Euler Elasticae......Page 185
References......Page 189
Part II Geometry as a Tool......Page 192
Contents......Page 193
6.1 Introduction......Page 194
6.2 Preliminaries on Variational Methods......Page 195
6.2.1 Exact Geometric Rod Theory via Variational Principles......Page 198
6.3.1 Configuration Manifold for Flexible Tubes Conveying Fluid......Page 202
6.3.2 Definition of the Lagrangian......Page 204
6.3.3 Variational Principle and Equations of Motion......Page 207
6.3.4 Incompressible Fluids......Page 210
6.3.5 Comparison with Previous Models......Page 211
6.3.6 Conservation Laws for Gas Motion and Rankine–Hugoniot Conditions......Page 212
6.4.1 Spatial Discretization......Page 214
6.4.2 Variational Integrator in Space and Time......Page 217
6.5 Further Developments......Page 220
References......Page 221
Contents......Page 224
7.1.1 Curvature-Dependent Functionals and Regularization......Page 225
7.1.2 Convex Relaxation of Curvature Regularization Functionals......Page 226
7.2.1 Concepts for Curve- (and Surface-) Lifting......Page 228
7.2.2 The Curvature Varifold Approach......Page 230
7.2.3 The Hyper-Varifold Approach......Page 232
7.2.4 The Gauss Graph Current Approach......Page 233
7.2.5 The Jump Set Calibration Approach......Page 235
7.3 Discretization Strategies......Page 237
7.3.1 Finite Differences......Page 238
7.3.3 Raviart–Thomas Finite Elements on a Staggered Grid......Page 239
7.3.4 Adaptive Line Measure Segments......Page 240
7.4 The Jump Set Calibration Approach in 3D......Page 241
7.4.1 Regularization Model......Page 242
7.4.2 Derivation of Theorem 7.21......Page 244
7.4.3 Adaptive Discretization with Surface Measures......Page 246
References......Page 248
Contents......Page 251
8.1 Introduction......Page 252
8.2.1.1 Dually Flat Statistical Manifolds......Page 254
8.2.1.2 The Assignment Manifold......Page 255
8.2.2.1 Likelihood Map......Page 258
8.2.2.2 Similarity Map......Page 259
8.2.2.3 Assignment Flow......Page 260
8.2.2.4 Geometric Integration......Page 262
8.2.2.5 Evaluation of Discrete Graphical Models: Wasserstein Message Passing......Page 263
8.2.2.6 Global Static vs. Local Dynamically Interacting Statistical Models......Page 264
8.3.1 Unsupervised Assignment Flow: Label Evolution......Page 266
8.3.2 Self-assignment Flow: Learning Labels from Data......Page 268
8.4.1 Linear Assignment Flow......Page 270
8.4.2 Parameter Estimation and Prediction......Page 271
8.5 Outlook......Page 273
References......Page 274
Contents......Page 277
9.1 Introduction......Page 278
9.1.1 Aims and Outline......Page 279
9.2 The Geometry of Low-Rank Matrices......Page 280
9.2.1 Singular Value Decomposition and Low-Rank Approximation......Page 281
9.2.2 Fixed Rank Manifold......Page 283
9.2.3 Tangent Space......Page 284
9.2.4 Retraction......Page 286
9.3 The Geometry of the Low-Rank Tensor Train Decomposition......Page 287
9.3.1 The Tensor Train Decomposition......Page 289
9.3.2 TT-SVD and Quasi Optimal Rank Truncation......Page 292
9.3.3 Manifold Structure......Page 295
9.3.4 Tangent Space and Retraction......Page 297
9.3.5 Elementary Operations and TT Matrix Format......Page 299
9.4.1 Riemannian Optimization......Page 302
9.4.2 Linear Systems......Page 305
9.4.3 Computational Cost......Page 306
9.4.4 Difference to Iterative Thresholding Methods......Page 307
9.4.5 Convergence......Page 309
9.4.6 Eigenvalue Problems......Page 310
9.5 Initial Value Problems......Page 311
9.5.1 Dynamical Low-Rank Approximation......Page 312
9.5.2 Approximation Properties......Page 313
9.5.3 Low-Dimensional Evolution Equations......Page 314
9.5.4 Projector-Splitting Integrator......Page 315
9.6 Applications......Page 318
9.6.1 Matrix Equations......Page 319
9.6.2 Schrödinger Equation......Page 320
9.6.4 Stochastic and Parametric Equations......Page 322
9.6.5 Transport Equations......Page 323
References......Page 324
Part III Statistical Methods and Non-linear Geometry......Page 330
Contents......Page 331
10.2 Some Euclidean Statistics Building on Mean and Covariance......Page 332
10.3 Fréchet ρ-Means and Their Strong Laws......Page 334
10.4 Procrustes Analysis Viewed Through Fréchet Means......Page 337
10.5 A CLT for Fréchet ρ-Means......Page 339
10.6 Geodesic Principal Component Analysis......Page 342
10.7 Backward Nested Descriptors Analysis (BNDA)......Page 344
10.8 Two Bootstrap Two-Sample Tests......Page 346
10.9 Examples of BNDA......Page 348
References......Page 350
Contents......Page 353
11.1 Introduction......Page 354
11.2 Means on Manifolds......Page 355
11.3 Statistics Beyond the Mean Value: Generalizing PCA......Page 358
11.3.1 Barycentric Subspaces in Manifolds......Page 360
11.3.2 From PCA to Barycentric Subspace Analysis......Page 363
11.3.3 Sample-Limited Lp Barycentric Subspace Inference......Page 365
11.4.1 Example on Synthetic Data in a Constant Curvature Space......Page 367
11.4.2 A Symmetric Group-Wise Analysis of Cardiac Motion in 4D Image Sequences......Page 368
11.5 Conclusion and Perspectives......Page 370
References......Page 371
12.1 Variational Inference......Page 374
12.1.1 Score Gradient......Page 377
12.1.2 Reparametrization Gradient......Page 379
12.2.1 Autoencoder......Page 381
12.2.2 Variational Autoencoder......Page 382
12.3 Generative Flows......Page 384
12.4 Geometric Variational Inference......Page 385
References......Page 387
Part IV Shapes Spaces and the Analysis of Geometric Data......Page 390
Contents......Page 391
13.1 Introduction......Page 392
13.2 Registration Problem and Elastic Framework......Page 393
13.2.1 The Use of the L2 Norm and Its Limitations......Page 394
13.2.2 Elastic Registration of Scalar Functions......Page 396
13.2.3.1 Registration of Curves......Page 398
13.2.3.2 Shape Space......Page 399
13.3 Shape Summary Statistics, Principal Modes and Models......Page 402
13.4 Conclusion......Page 404
References......Page 405
Contents......Page 407
14.1 Introduction and Background......Page 408
14.2 Elastic Shape Analysis of Open Curves......Page 411
14.3 Elastic Analysis of Trajectories......Page 412
14.4 Joint Framework of Analyzing Shapes and Trajectories......Page 413
14.4.1 Trajectories of Functions......Page 414
14.4.2 Trajectories of Tensors......Page 420
14.5 Conclusion......Page 422
References......Page 423
Contents......Page 426
15.1 Introduction......Page 427
15.2 Background and Definitions......Page 428
15.3 Topological Feature Representations......Page 432
15.4 Geometric Metrics for Representations......Page 437
15.5.1 Time-Series Analysis......Page 440
15.5.2 Image Analysis......Page 441
15.5.3 Shape Analysis......Page 442
References......Page 446
Contents......Page 453
16.1 Introduction......Page 454
16.2.1 Geometric Invariants of Curves......Page 455
16.2.1.1 Differential Invariants......Page 456
16.2.1.2 Integral Invariants......Page 458
16.2.2.2 Training for Invariance......Page 460
16.2.2.3 Results......Page 462
16.3.1 Geometric Moments as Class Identifiers......Page 464
16.3.2 Raw Point Cloud Classification Based on Moments......Page 465
Performance Evaluation......Page 467
References......Page 469
Contents......Page 472
17.1 Introduction......Page 473
17.2.1 Spaces of Plane Curves......Page 474
17.2.2 Basic Sub-Riemannian Structure......Page 475
17.2.3 Generalization......Page 478
17.2.4 Pontryagin's Maximum Principle......Page 482
17.3.1 Control Points......Page 484
17.3.2 Scale Attributes......Page 487
17.4.1 Definition......Page 489
17.4.2 Basic Deformation Modules......Page 491
17.4.3 Simple Matching Example......Page 492
17.4.4 Population Analysis......Page 495
17.5 Constrained Evolution......Page 497
17.6 Conclusion......Page 500
References......Page 501
Part V Optimization Algorithms and Numerical Methods......Page 505
Contents......Page 506
18.1 Introduction......Page 507
18.2 Notations and Basic Results......Page 508
18.3 Examples of Convex Functions on Riemannian Manifolds......Page 510
18.3.2 Example in the Euclidean Space with a New Riemannian Metric......Page 511
18.3.3 Examples in the Positive Orthant with a New Riemannian Metric......Page 512
Bibliographic Notes and Remarks......Page 513
18.4 Gradient Method for Optimization......Page 514
18.4.1 Asymptotic Convergence Analysis......Page 515
18.4.2 Iteration-Complexity Analysis......Page 517
Bibliographic Notes and Remarks......Page 518
18.5 Subgradient Method for Optimization......Page 519
18.5.1 Asymptotic Convergence Analysis......Page 520
18.5.2 Iteration-Complexity Analysis......Page 523
Bibliographic Notes and Remarks......Page 524
18.6 Proximal Point Method for Optimization......Page 525
18.6.2 Iteration-Complexity Analysis......Page 526
Bibliographic Notes and Remarks......Page 527
References......Page 528
Contents......Page 533
19.1 Introduction......Page 534
Additional Background and Summary......Page 535
19.2 Key Definitions......Page 536
19.3 Stochastic Gradient Descent on Manifolds......Page 540
19.4 Accelerating Stochastic Gradient Descent......Page 544
19.5 Analysis for G-Convex and Gradient Dominated Functions......Page 548
19.6 Example Applications......Page 555
References......Page 558
Contents......Page 561
20.1 Introduction......Page 562
20.2 ALM Properties......Page 563
20.3 Geodesic Distance Based Averaging Techniques......Page 564
20.3.1 Karcher Mean (L2 Riemannian Mean)......Page 565
20.3.2 Riemannian Median (L1 Riemannian Mean)......Page 566
20.4 Divergence-Based Averaging Techniques......Page 568
20.4.1.1 The α-Divergence Family......Page 569
20.4.1.3 The LogDet α-Divergence......Page 570
20.4.1.4 The LogDet Bregman Divergence......Page 571
20.4.1.6 The von Neumann Bregman Divergence......Page 572
20.4.2 Left, Right, and Symmetrized Means Using Divergences......Page 573
20.4.2.2 The LogDet Bregman Divergence......Page 574
20.4.2.3 The von Neumann Bregman Divergence......Page 575
20.5 Alternative Metrics on SPD Matrices......Page 576
20.6 Conclusion......Page 577
References......Page 578
Contents......Page 582
21.1 Introduction......Page 583
21.2.1 Mathematical Setting......Page 584
21.2.2 Rolling Manifolds......Page 587
21.2.3 Parallel Transport......Page 596
21.3.1 Formulation of the Problem......Page 597
21.3.2 Motivation......Page 598
21.3.3 Solving the Interpolation Problem......Page 599
21.3.5 Implementation of the Algorithm on S2......Page 601
21.4.1 Rolling a Hypersurface......Page 605
21.4.2 The Case of an Ellipsoid......Page 607
21.4.2.1 Changing the Metric on the Ellipsoid......Page 608
21.4.2.2 Implementation of the Interpolation Algorithm on E2......Page 609
21.4.3.1 Rolling Symmetric Spaces......Page 610
21.4.3.6 Variational and Hamiltonian Approach to Interpolation......Page 611
21.4.3.8 Some Applications......Page 612
References......Page 613
Part VI Applications......Page 616
Contents......Page 617
22.1.1 Motivation......Page 618
22.1.2 General Model......Page 619
22.2.1 Problem-Specific Manifold and Model......Page 621
22.2.2 Numerical Approach......Page 624
22.2.3 Experiments......Page 626
22.2.4 Discussion......Page 628
22.3.1 Problem-specific Manifold and Model......Page 630
22.3.2 Algorithmic Approach......Page 633
22.3.3 Experiments......Page 635
22.3.4 Discussion......Page 637
22.4.1 Problem-Specific Manifold, Basic Model and Algorithm......Page 640
22.4.2 Experiments......Page 643
22.4.3 Extensions......Page 644
22.5 Conclusion and Outlook......Page 646
References......Page 647
Contents......Page 652
23.1 Landmark Representation......Page 653
23.1.1 Challenges......Page 654
23.2 Static Representation......Page 655
23.3.1 Mathematical Preliminaries......Page 656
23.3.1.1 Grassmann Manifold......Page 657
23.3.2 Riemannian Manifold of Positive Semi-Definite Matrices of Fixed Rank......Page 658
23.3.2.2 Pseudo-Geodesics and Closeness in S+(d,n)......Page 659
23.4 Gram Matrix Trajectories for Temporal Modeling of Landmark Sequences......Page 660
23.4.1 Rate-Invariant Comparison of Gram Matrix Trajectories......Page 661
23.5.1 Pairwise Proximity Function SVM......Page 662
23.6.1.2 Experimental Settings and Parameters......Page 663
23.6.1.3 Results and Discussion......Page 664
23.6.2.1 Datasets......Page 665
23.6.2.3 Results and Discussion......Page 666
23.7 Affine-Invariant Shape Representation Using Barycentric Coordinates......Page 667
23.7.1 Relationship with the Conventional Grassmannian Representation......Page 669
23.8 Metric Learning on Barycentric Representation for Expression Recognition in Unconstrained Environments......Page 670
23.8.1 Experimental Results......Page 671
23.8.1.1 Results and Discussions......Page 672
References......Page 673
Contents......Page 677
24.1 Introduction......Page 678
24.2.1 Probability Density Functions......Page 680
24.2.2 Amplitude and Phase in Elastic Functional Data......Page 682
24.2.3 Shapes of Open and Closed Curves......Page 683
24.2.4 Shapes of Surfaces......Page 685
24.3.1 Karcher Mean......Page 686
24.3.2 Covariance Estimation and Principal Component Analysis......Page 687
24.4.1 Probability Density Functions......Page 688
24.4.2 Amplitude and Phase in Elastic Functional Data......Page 690
24.4.3 Shapes of Open and Closed Curves......Page 694
24.4.4 Shapes of Surfaces......Page 697
24.5 Summary......Page 699
References......Page 700