Mathematical Principles of Topological and Geometric Data Analysis (Mathematics of Data, 2)

✍ Scribed by Parvaneh Joharinad, Jürgen Jost

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 287
• Category: Library

✦ Synopsis

This book explores and demonstrates how geometric tools can be used in data analysis. Beginning with a systematic exposition of the mathematical prerequisites, covering topics ranging from category theory to algebraic topology, Riemannian geometry, operator theory and network analysis, it goes on to describe and analyze some of the most important machine learning techniques for dimension reduction, including the different types of manifold learning and kernel methods. It also develops a new notion of curvature of generalized metric spaces, based on the notion of hyperconvexity, which can be used for the topological representation of geometric information.

In recent years there has been a fascinating development: concepts and methods originally created in the context of research in pure mathematics, and in particular in geometry, have become powerful tools in machine learning for the analysis of data. The underlying reason for this is that data are typically equipped with somekind of notion of distance, quantifying the differences between data points. Of course, to be successfully applied, the geometric tools usually need to be redefined, generalized, or extended appropriately.

Primarily aimed at mathematicians seeking an overview of the geometric concepts and methods that are useful for data analysis, the book will also be of interest to researchers in machine learning and data analysis who want to see a systematic mathematical foundation of the methods that they use.

✦ Table of Contents

Preface
Contents
1 Introduction
1.1 Examples of Data
1.2 Strategies
1.3 Qualitative and Quantitative Properties
1.4 Generalization and Distinction
1.5 Prior Structural Assumptions
1.6 Geometric Machine Learning
2 Topological Foundations, Hypercomplexes and Homology
2.1 Simplicial Complexes and Their Homology
2.2 Hypergraphs
2.3 Category Theory
2.3.1 Categories and Functors
2.3.2 Diagrams
2.3.3 Presheaves
2.3.4 Fuzzy Simplicial Sets
2.4 Topological Concepts
2.4.1 Topological Spaces
2.4.2 Coverings
2.4.3 Singular Homology
2.4.4 Manifolds
2.5 Measures
2.6 Persistent Homology
2.7 Principles of Homology Theory
2.7.1 +1-1=0, or Cancellations and the EulerCharacteristic
2.7.2 a-b=(a-c)-(b-c), or Relative Homology and the Excision Theorem
2.7.3 a=(a-a1)+(a1-a2)+…+ an, or Filtrations
3 Weighted Complexes, Cohomology and Laplace Operators
3.1 Cohomology
3.1.1 The Cohomology Ring
3.2 Eigenvalues of Linear Operators
3.3 Laplace Operators
4 The Laplace Operator and the Geometry of Graphs
4.1 Scalar Products and Laplace Operators on Graphs
4.2 The Algebraic Graph Laplacian
4.3 The (Normalized) Laplacian and Its Spectrum on Graphs
4.4 Random Walks and Iterated Graphs
4.4.1 Random Walks
4.4.2 Iterated Graphs
4.4.3 The Laplacian of Iterated Graphs
5 Metric Spaces and Manifolds
5.1 Metric Spaces
5.2 Riemannian Manifolds
5.2.1 Differentiable Structures and Tensors
5.2.2 Riemannian Metrics
5.2.3 Orientation and Integration on Riemannian Manifold
5.2.4 Geodesics and the Exponential Map
5.2.5 The Laplace Operator and the Heat Equation
5.2.6 Curvature
5.2.7 Ricci Curvature
5.2.8 Jacobi Fields and Integration in Polar Coordinates
5.3 Barycenters and Averaging
5.4 L1- and L∞-Spaces
5.5 The Largitude of Metric Spaces
5.5.1 Maximally Distant Points
5.5.2 Computational Aspects
5.5.3 Graphs
5.5.4 Riemannian Manifolds
5.5.5 Largitude
6 Linear Methods: Kernels, Variations, and Averaging
6.1 Geometry Underlying Data Analysis
6.2 Some Terminological Preliminaries
6.3 Kernels
6.4 Principal Component Analysis
6.5 Averaging
6.5.1 Laplace Operators and Harmonic Functions
6.5.2 An Application to Image Denoising
6.5.3 Harmonic Functions and Mappings in the Context of Metric Geometry
7 Nonlinear Schemes: Clustering, Feature Extraction and Dimension Reduction
7.1 Vector Quantization, Data Compression and Factor Analysis
7.2 The Kohonen Algorithm as an Example of a Nonlinear Scheme
7.2.1 Definition and Analysis of the Kohonen Algorithm
7.3 Non-linear Dimension Reduction Schemes
7.3.1 Isomap
7.3.2 Local Linear Embedding
7.3.3 Laplacian Eigenmap
7.3.4 Diffusion Map
7.3.5 t-Stochastic Neighborhood Embedding (t-SNE)
7.3.6 Uniform Manifold Approximation andProjection (UMAP)
7.4 Implementation
8 Manifold Learning, the Scheme of Laplacian Eigenmaps
8.1 Eigenfunctions and Embeddings
8.2 Random Walks on Manifolds
8.3 Manifolds and Graphs
8.4 Green's Function
9 Metrics and Curvature
9.1 Hyperconvexity
9.2 Injective Metric Spaces
9.3 Curvature of Metric Spaces
9.4 Translating Geometry into Topology
Bibliography
Index