How to reveal, characterize, and exploit the structure in data? Meeting this central
challenge of modern data science requires the development of new mathematical approaches to data analysis, going beyond traditional statistical methods.
Fruitful mathematical methods can originate in geometry, topology, algebra, analysis, stochastics, combinatorics, or indeed virtually any field of mathematics.
Confronting the challenge of structure in data is already leading to productive
new interactions among mathematics, statistics, and computer science, notably in
machine learning. We invite novel contributions (research monographs, advanced
textbooks, and lecture notes) presenting substantial mathematics that is relevant
for data science. Since the methods required to understand data depend on the
source and type of the data, we very much welcome contributions comprising
significant discussions of the problems presented by particular applications. We
also encourage the use of online resources for exercises, software and data sets.
Contributions from all mathematical communities that analyze structures in data
are welcome. Examples of potential topics include optimization, topological data
analysis, compressed sensing, algebraic statistics, information geometry, manifold
learning, tensor decomposition, support vector machines, neural networks, and
many more.
Author(s): Hal Schenck
Series: Mathematics of Data, 1
Publisher: Springer
Year: 2022
Language: English
Pages: 230
City: Cham
Preface
Contents
1 Linear Algebra Tools for Data Analysis
1.1 Linear Equations, Gaussian Elimination, Matrix Algebra
1.2 Vector Spaces, Linear Transformations, Basis and Change of Basis
1.2.1 Basis of a Vector Space
1.2.2 Linear Transformations
1.2.3 Change of Basis
1.3 Diagonalization, Webpage Ranking, Data and Covariance
1.3.1 Eigenvalues and Eigenvectors
1.3.2 Diagonalization
1.3.3 Ranking Using Diagonalization
1.3.4 Data Application: Diagonalization of the Covariance Matrix
1.4 Orthogonality, Least Squares Fitting, Singular Value Decomposition
1.4.1 Least Squares
1.4.2 Subspaces and Orthogonality
1.4.3 Singular Value Decomposition
2 Basics of Algebra: Groups, Rings, Modules
2.1 Groups, Rings and Homomorphisms
2.1.1 Groups
2.1.2 Rings
2.2 Modules and Operations on Modules
2.2.1 Ideals
2.2.2 Tensor Product
2.2.3 Hom
2.3 Localization of Rings and Modules
2.4 Noetherian Rings, Hilbert Basis Theorem, Varieties
2.4.1 Noetherian Rings
2.4.2 Solutions to a Polynomial System: Varieties
3 Basics of Topology: Spaces and Sheaves
3.1 Topological Spaces
3.1.1 Set Theory and Equivalence Relations
3.1.2 Definition of a Topology
3.1.3 Discrete, Product, and Quotient Topologies
3.2 Vector Bundles
3.3 Sheaf Theory
3.3.1 Presheaves and Sheaves
3.3.2 Posets, Direct Limit, and Stalks
3.3.3 Morphisms of Sheaves and Exactness
3.4 From Graphs to Social Media to Sheaves
3.4.1 Spectral Graph Theory
3.4.2 Heat Diffusing on a Wire Graph
3.4.3 From Graphs to Cellular Sheaves
4 Homology I: Simplicial Complexes to Sensor Networks
4.1 Simplicial Complexes, Nerve of a Cover
4.1.1 The Nerve of a Cover
4.2 Simplicial and Singular Homology
4.2.1 Singular homology
4.3 Snake Lemma and Long Exact Sequence in Homology
4.3.1 Maps of complexes, Snake Lemma
4.3.2 Chain Homotopy
4.4 Mayer–Vietoris, Rips and Čech Complex, Sensor Networks
4.4.1 Mayer–Vietoris Sequence
4.4.2 Relative Homology
4.4.3 Čech Complex and Rips Complex
5 Homology II: Cohomology to Ranking Problems
5.1 Cohomology: Simplicial, Čech, de Rham Theories
5.1.1 Simplicial Cohomology
5.1.2 Čech Cohomology
5.1.3 de Rham Cohomology
5.2 Ranking, the Netflix Problem, and Hodge Theory
5.2.1 Hodge Decomposition
5.2.2 Application to Ranking
5.3 CW Complexes and Cellular Homology
5.4 Poincaré and Alexander Duality: Sensor Networks Revisited
5.4.1 Statement of Theorems and Examples
5.4.2 Alexander Duality: Proof
5.4.3 Sensor Networks Revisited
5.4.4 Poincaré Duality
6 Persistent Algebra: Modules Over a PID
6.1 Principal Ideal Domains and Euclidean Domains
6.2 Rational Canonical Form of a Matrix
6.3 Linear Transformations, K[t]-Modules, Jordan Form
6.4 Structure of Abelian Groups and Persistent Homology
6.4.1 Z-Graded Rings
7 Persistent Homology
7.1 Barcodes, Persistence Diagrams, Bottleneck Distance
7.1.1 History
7.1.2 Persistent Homology and the Barcode
7.1.3 Computation of Persistent Homology
7.1.4 Alpha and Witness Complexes
7.1.5 Persistence Diagrams
7.1.6 Metrics on Diagrams
7.2 Morse Theory
7.3 The Stability Theorem
7.4 Interleaving and Categories
7.4.1 Categories and Functors
7.4.2 Interleaving
7.4.3 Interleaving Vignette: Merge Trees
7.4.4 Zigzag Persistence and Quivers
8 Multiparameter Persistent Homology
8.1 Definition and Examples
8.1.1 Multiparameter Persistence
8.2 Graded Algebra, Hilbert Function, Series, Polynomial
8.2.1 The Hilbert Function
8.2.2 The Hilbert Series
8.3 Associated Primes and Zn-Graded Modules
8.3.1 Geometry of Sheaves
8.3.2 Associated Primes and Primary Decomposition
8.3.3 Additional Structure in the Zn-Graded Setting
8.4 Filtrations and Ext
9 Derived Functors and Spectral Sequences
9.1 Injective and Projective Objects, Resolutions
9.1.1 Projective and Injective Objects
9.1.2 Resolutions
9.2 Derived Functors
9.2.1 Categories and Functors
9.2.2 Constructing Derived Functors
9.2.3 Ext
9.2.4 The Global Sections Functor
9.2.5 Acyclic Objects
9.3 Spectral Sequences
9.3.1 Total Complex of Double Complex
9.3.2 The Vertical Filtration
9.3.3 Main Theorem
9.4 Pas de Deux: Spectral Sequences and Derived Functors
9.4.1 Resolution of a Complex
9.4.2 Grothendieck Spectral Sequence
9.4.3 Comparing Cohomology Theories
9.4.4 Cartan–Eilenberg Resolution
A Examples of Software Packages
A.1 Covariance and Spread of Data via R
A.2 Persistent Homology via scikit-tda
A.3 Computational Algebra via Macaulay2
A.4 Multiparameter Persistence via RIVET
Bibliography
Index
