Front matter
Copyright
Contents
Preface
Prelude
1 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
2 Complexes and Homology Groups
2.1 Simplicial Complex
2.2 Nerves, ห 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
2.5.3 Singular Homology
2.5.4 Cohomology
2.6 Notes and Exercises
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
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
5 Generators and Optimality
5.1 Optimal Generators/Basis
5.1.1 Greedy Algorithm for Optimal Hp(K )-Basis
5.1.2 Optimal H1(K )-Basis and Independence Check
5.2 Localization
5.2.1 Linear Program
5.2.2 Total Unimodularity
5.2.3 Relative Torsion
5.3 Persistent Cycles
5.3.1 Finite Intervals for Weak (p + 1)-Pseudomanifolds
5.3.2 Algorithm Correctness
5.3.3 Infinite Intervals for Weak (p + 1)- Pseudomanifolds Embedded in Rp+1
5.4 Notes and Exercises
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
7 Reeb Graphs
7.1 Reeb Graph: Definitions and Properties
7.2 Algorithms in the PL-Setting
7.2.1 An O(m log m)-Time Algorithm via Dynamic Graph Connectivity
7.2.2 A Randomized Algorithm with O(m log m) 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
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
9 Cover, Nerve, and Mapper
9.1 Covers and Nerves
9.1.1 Special Case of H1
9.2 Analysis of Persistent H1-Classes
9.3 Mapper and Multiscale Mapper
9.3.1 Multiscale Mapper
9.3.2 Persistence of H1-Classes in Mapper and Multiscale Mapper
9.4 Stability
9.4.1 Interleaving of Cover Towers and Multiscale Mappers
9.4.2 (c, s)-Good Covers
9.4.3 Relation to Intrinsic ห Cech 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
10 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
10.4.3 Theoretical Guarantees
10.5 Applications
10.5.1 Road Network
10.5.2 Neuron Network
10.6 Notes and Exercises
11 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 d-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
12 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 dI for Intervals
12.5 Notes and Exercises
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
References
Index