Mathematics of Big Data Spreadsheets, Databases, Matrices, and Graphs

✍ Scribed by Jeremy Kepner and Hayden Jananthan

• Publisher: The MIT Press
• Year: 2018
• Tongue: English
• Leaves: 490
• Category: Library

✦ Table of Contents

Preface
About the Authors
About the Cover
Acknowledgments
Applications and Practice
Introduction and Overview
Mathematics of Data
Data in the World
Mathematical Foundations
Making Data Rigorous
Conclusions, Exercises, and References
Perspectives on Data
Interrelations
Spreadsheets
Databases
Matrices
Graphs
Map Reduce
Other Perspectives
Conclusions, Exercises, and References
Dynamic Distributed Dimensional Data Model
Background
Design
Matrix Mathematics
Common SQL, NoSQL, NewSQL Interface
Key-Value Store Database Schema
Data-Independent Analytics
Parallel Performance
Computing on Masked Data
Conclusions, Exercises, and References
Associative Arrays and Musical Metadata
Data and Metadata
Dense Data
Dense Operations
Sparse Data
Sparse Operations
Conclusions, Exercises, and References
Associative Arrays and Abstract Art
Visual Abstraction
Minimal Adjacency Array
Symmetric Adjacency Array
Weighted Adjacency Array
Incidence Array
Conclusions, Exercises, and References
Manipulating Graphs with Matrices
Introduction
Matrix Indices and Values
Composable Graph Operations and Linear Systems
Matrix Graph Operations Overview
Graph Algorithms and Diverse Semirings
Conclusions, Exercises, and References
Graph Analysis and Machine Learning Systems
Introduction
Data Representation
Graph Construction
Adjacency Array Graph Traversal
Incidence Array Graph Traversal
Vertex Degree Centrality
Edge Degree Centrality
Eigenvector Centrality
Singular Value Decomposition
PageRank
Deep Neural Networks
Conclusions, Exercises, and References
Mathematical Foundations
Visualizing the Algebra of Associative Arrays
Associative Array Analogs of Matrix Operations
Abstract Algebra for Computer Scientists and Engineers
Depicting Mathematics
Associative Array Class Diagrams
Set
Semiring
Linear Algebra
Ordered Sets
Boolean Algebra
Associative Array Algebra
Conclusions, Exercises, and References
Defining the Algebra of Associative Arrays
Operations on Sets
Ordered Sets
Supremum and Infimum
Lattice
The Semirings of Interest
Conclusions, Exercises, and References
Structural Properties of Associative Arrays
Estimating Structure
Associative Array Formal Definition
Padding Associative Arrays with Zeros
Zero, Null, Zero-Sum-Free
Properties of Matrices and Associative Arrays
Properties of Zero Padding
Support and Size
Image and Rank
Example: Music
Example: Art
Properties of Element-Wise Addition
Properties of Element-Wise Multiplication
Array Multiplication
Closure of Operations between Arrays
Conclusions, Exercises, and References
Graph Construction and Graphical Patterns
Introduction
Adjacency and Incidence Array Definitions
Adjacency Array Construction
Graph Construction with Different Semirings
Special Arrays and Graphs
Key Ordering
Algebraic Properties
Subobject Properties
Conclusions, Exercises, and References
Linear Systems
Survey of Common Transformations
Array Transformations
Identity
Contraction
Stretching
Rotation
Conclusions, Exercises, and References
Maps and Bases
Semimodules
Linear Maps
Linear Independence and Bases
Existence of Bases
Size of Bases
Semialgebras and the Algebra of Arrays
Conclusions, Exercises, and References
Linearity of Associative Arrays
The Null Space of Linear Maps
Supremum-Blank Algebras
Max-Blank Structure Theorem
Examples of Supremum-Blank Algebras
Explicit Computations of x(A,w) for Supremum-Blank Algebras
Conclusions, Exercises, and References
Eigenvalues and Eigenvectors
Introduction
Quasi-Inverses
Existence of Eigenvalues for Idempotent Multiplication
Strong Dependence and Characteristic Bipolynomial
Eigenanalysis for Irreducible Matrices for Invertible Multiplication
Eigen-Semimodules
Singular Value Decomposition
Conclusions, Exercises, and References
Higher Dimensions
d-Dimensional Associative Arrays
Key Ordering and Two-Dimensional Projections
Algebraic Properties
Subarray Properties
Conclusions, Exercises, and References
Appendix: Notation
Index
Answers to Selected Exercises