Computational Topology for Data Analysis 1st edition by Tamal Krishna Dey, Yussu Wang ISBN ‎ 1009098160 ‎ 978-1009098168

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ISBN 10: ‎ 1009098160
ISBN 13: ‎ 978-1009098168
Author: Tamal Krishna Dey, Yussu Wang

Topological data analysis (TDA) has emerged recently as a viable tool for analyzing complex data, and the area has grown substantially both in its methodologies and applicability. Providing a computational and algorithmic foundation for techniques in TDA, this comprehensive, self-contained text introduces students and researchers in mathematics and computer science to the current state of the field. The book features a description of mathematical objects and constructs behind recent advances, the algorithms involved, computational considerations, as well as examples of topological structures or ideas that can be used in applications. It provides a thorough treatment of persistent homology together with various extensions – like zigzag persistence and multiparameter persistence – and their applications to different types of data, like point clouds, triangulations, or graph data. Other important topics covered include discrete Morse theory, the Mapper structure, optimal generating cycles, as well as recent advances in embedding TDA within machine learning frameworks.

Computational Topology for Data Analysis 1st Table of contents:

1 Basics

1.1 Topological Space

1.2 Metric Space Topology

1.3 Maps, Homeomorphisms, and Homotopies

1.4 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, Čech 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 H[sub(p)](K)-Basis

5.1.2 Optimal H[sub(1)](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 forWeak (p + 1)-Pseudomanifolds

5.3.2 Algorithm Correctness

5.3.3 Infinite Intervals for Weak (p + 1)-Pseudomanifolds Embedded in R[sup(p+1)]

5.4 Notes and Exercises

6 Topological Analysis of Point Clouds

6.1 Persistence for Rips and Čech 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 logm)-Time Algorithm via Dynamic Graph Connectivity

7.2.2 A Randomized Algorithm with O(m logm) 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 H[sub(1)]

9.2 Analysis of Persistent H[sub(1)]-Classes

9.3 Mapper and Multiscale Mapper

9.3.1 Multiscale Mapper

9.3.2 Persistence of H[sub(1)]-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 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

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

11.6 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 d[sub(I)] 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

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Tags: Tamal Krishna Dey, Yussu Wang, Computational Topology, Data Analysis