Essential Math for AI (Final Version)

Cover
Copyright
Table of Contents
Preface
	Why I Wrote This Book
	Who Is This Book For?
	Who Is This Book Not For?
	How Will the Math Be Presented in This Book?
	Infographic
	What Math Background Is Expected from You to Be Able to Read This Book?
	Overview of the Chapters
	My Favorite Books on AI
	Conventions Used in This Book
	Using Code Examples
	O’Reilly Online Learning
	How to Contact Us
	Acknowledgments
Chapter 1. Why Learn the Mathematics of AI?
	What Is AI?
	Why Is AI So Popular Now?
	What Is AI Able to Do?
		An AI Agent’s Specific Tasks
	What Are AI’s Limitations?
	What Happens When AI Systems Fail?
	Where Is AI Headed?
	Who Are the Current Main Contributors to the AI Field?
	What Math Is Typically Involved in AI?
	Summary and Looking Ahead
Chapter 2. Data, Data, Data
	Data for AI
	Real Data Versus Simulated Data
	Mathematical Models: Linear Versus Nonlinear
	An Example of Real Data
	An Example of Simulated Data
	Mathematical Models: Simulations and AI
	Where Do We Get Our Data From?
	The Vocabulary of Data Distributions, Probability, and Statistics
		Random Variables
		Probability Distributions
		Marginal Probabilities
		The Uniform and the Normal Distributions
		Conditional Probabilities and Bayes’ Theorem
		Conditional Probabilities and Joint Distributions
		Prior Distribution, Posterior Distribution, and Likelihood Function
		Mixtures of Distributions
		Sums and Products of Random Variables
		Using Graphs to Represent Joint Probability Distributions
		Expectation, Mean, Variance, and Uncertainty
		Covariance and Correlation
		Markov Process
		Normalizing, Scaling, and/or Standardizing a Random Variable or Data Set
		Common Examples
	Continuous Distributions Versus Discrete Distributions (Density Versus Mass)
	The Power of the Joint Probability Density Function
	Distribution of Data: The Uniform Distribution
	Distribution of Data: The Bell-Shaped Normal (Gaussian) Distribution
	Distribution of Data: Other Important and Commonly Used Distributions
	The Various Uses of the Word “Distribution”
	A/B Testing
	Summary and Looking Ahead
Chapter 3. Fitting Functions to Data
	Traditional and Very Useful Machine Learning Models
	Numerical Solutions Versus Analytical Solutions
	Regression: Predict a Numerical Value
		Training Function
		Loss Function
		Optimization
	Logistic Regression: Classify into Two Classes
		Training Function
		Loss Function
		Optimization
	Softmax Regression: Classify into Multiple Classes
		Training Function
		Loss Function
		Optimization
	Incorporating These Models into the Last Layer of a Neural Network
	Other Popular Machine Learning Techniques and Ensembles of Techniques
		Support Vector Machines
		Decision Trees
		Random Forests
		k-means Clustering
	Performance Measures for Classification Models
	Summary and Looking Ahead
Chapter 4. Optimization for Neural Networks
	The Brain Cortex and Artificial Neural Networks
	Training Function: Fully Connected, or Dense, Feed Forward Neural Networks
		A Neural Network Is a Computational Graph Representation of the Training Function
		Linearly Combine, Add Bias, Then Activate
		Common Activation Functions
		Universal Function Approximation
		Approximation Theory for Deep Learning
	Loss Functions
	Optimization
		Mathematics and the Mysterious Success of Neural Networks
		Gradient Descent       ω → i+1     =    ω → i     -    η    ∇    L          (      ω → i       )
		Explaining the Role of the Learning Rate Hyperparameter   η
		Convex Versus Nonconvex Landscapes
		Stochastic Gradient Descent
		Initializing the Weights   ω → 0  for the Optimization Process
	Regularization Techniques
		Dropout
		Early Stopping
		Batch Normalization of Each Layer
		Control the Size of the Weights by Penalizing Their Norm
		Penalizing the   l 2  Norm Versus Penalizing the   l 1  Norm
		Explaining the Role of the Regularization Hyperparameter   α
	Hyperparameter Examples That Appear in Machine Learning
	Chain Rule and Backpropagation: Calculating       ∇    L    (    ω → i     )
		Backpropagation Is Not Too Different from How Our Brain Learns
		Why Is It Better to Backpropagate?
		Backpropagation in Detail
	Assessing the Significance of the Input Data Features
	Summary and Looking Ahead
Chapter 5. Convolutional Neural Networks and Computer Vision
	Convolution and Cross-Correlation
		Translation Invariance and Translation Equivariance
		Convolution in Usual Space Is a Product in Frequency Space
	Convolution from a Systems Design Perspective
		Convolution and Impulse Response for Linear and Translation Invariant Systems
	Convolution and One-Dimensional Discrete Signals
	Convolution and Two-Dimensional Discrete Signals
		Filtering Images
		Feature Maps
	Linear Algebra Notation
		The One-Dimensional Case: Multiplication by a Toeplitz Matrix
		The Two-Dimensional Case: Multiplication by a Doubly Block Circulant Matrix
	Pooling
	A Convolutional Neural Network for Image Classification
	Summary and Looking Ahead
Chapter 6. Singular Value Decomposition: Image Processing, Natural Language Processing, and Social Media
	Matrix Factorization
	Diagonal Matrices
	Matrices as Linear Transformations Acting on Space
		Action of A on the Right Singular Vectors
		Action of A on the Standard Unit Vectors and the Unit Square Determined by Them
		Action of A on the Unit Circle
		Breaking Down the Circle-to-Ellipse Transformation According to the Singular Value Decomposition
		Rotation and Reflection Matrices
		Action of A on a General Vector   x →
	Three Ways to Multiply Matrices
	The Big Picture
		The Condition Number and Computational Stability
	The Ingredients of the Singular Value Decomposition
	Singular Value Decomposition Versus the Eigenvalue Decomposition
	Computation of the Singular Value Decomposition
		Computing an Eigenvector Numerically
	The Pseudoinverse
	Applying the Singular Value Decomposition to Images
	Principal Component Analysis and Dimension Reduction
	Principal Component Analysis and Clustering
	A Social Media Application
	Latent Semantic Analysis
	Randomized Singular Value Decomposition
	Summary and Looking Ahead
Chapter 7. Natural Language and Finance AI: Vectorization and Time Series
	Natural Language AI
	Preparing Natural Language Data for Machine Processing
	Statistical Models and the log Function
	Zipf’s Law for Term Counts
	Various Vector Representations for Natural Language Documents
		Term Frequency Vector Representation of a Document or Bag of Words
		Term Frequency-Inverse Document Frequency Vector Representation of a Document
		Topic Vector Representation of a Document Determined by Latent Semantic Analysis
		Topic Vector Representation of a Document Determined by Latent Dirichlet Allocation
		Topic Vector Representation of a Document Determined by Latent Discriminant Analysis
		Meaning Vector Representations of Words and of Documents Determined by Neural Network Embeddings
	Cosine Similarity
	Natural Language Processing Applications
		Sentiment Analysis
		Spam Filter
		Search and Information Retrieval
		Machine Translation
		Image Captioning
		Chatbots
		Other Applications
	Transformers and Attention Models
		The Transformer Architecture
		The Attention Mechanism
		Transformers Are Far from Perfect
	Convolutional Neural Networks for Time Series Data
	Recurrent Neural Networks for Time Series Data
		How Do Recurrent Neural Networks Work?
		Gated Recurrent Units and Long Short-Term Memory Units
	An Example of Natural Language Data
	Finance AI
	Summary and Looking Ahead
Chapter 8. Probabilistic Generative Models
	What Are Generative Models Useful For?
	The Typical Mathematics of Generative Models
	Shifting Our Brain from Deterministic Thinking to Probabilistic Thinking
	Maximum Likelihood Estimation
	Explicit and Implicit Density Models
	Explicit Density-Tractable: Fully Visible Belief Networks
		Example: Generating Images via PixelCNN and Machine Audio via WaveNet
	Explicit Density-Tractable: Change of Variables Nonlinear Independent Component Analysis
	Explicit Density-Intractable: Variational Autoencoders Approximation via Variational Methods
	Explicit Density-Intractable: Boltzman Machine Approximation via Markov Chain
	Implicit Density-Markov Chain: Generative Stochastic Network
	Implicit Density-Direct: Generative Adversarial Networks
		How Do Generative Adversarial Networks Work?
	Example: Machine Learning and Generative Networks for High Energy Physics
	Other Generative Models
		Naive Bayes Classification Model
		Gaussian Mixture Model
	The Evolution of Generative Models
		Hopfield Nets
		Boltzmann Machine
		Restricted Boltzmann Machine (Explicit Density and Intractable)
		The Original Autoencoder
	Probabilistic Language Modeling
	Summary and Looking Ahead
Chapter 9. Graph Models
	Graphs: Nodes, Edges, and Features for Each
	Example: PageRank Algorithm
	Inverting Matrices Using Graphs
	Cayley Graphs of Groups: Pure Algebra and Parallel Computing
	Message Passing Within a Graph
	The Limitless Applications of Graphs
		Brain Networks
		Spread of Disease
		Spread of Information
		Detecting and Tracking Fake News Propagation
		Web-Scale Recommendation Systems
		Fighting Cancer
		Biochemical Graphs
		Molecular Graph Generation for Drug and Protein Structure Discovery
		Citation Networks
		Social Media Networks and Social Influence Prediction
		Sociological Structures
		Bayesian Networks
		Traffic Forecasting
		Logistics and Operations Research
		Language Models
		Graph Structure of the Web
		Automatically Analyzing Computer Programs
		Data Structures in Computer Science
		Load Balancing in Distributed Networks
		Artificial Neural Networks
	Random Walks on Graphs
	Node Representation Learning
	Tasks for Graph Neural Networks
		Node Classification
		Graph Classification
		Clustering and Community Detection
		Graph Generation
		Influence Maximization
		Link Prediction
	Dynamic Graph Models
	Bayesian Networks
		A Bayesian Network Represents a Compactified Conditional Probability Table
		Making Predictions Using a Bayesian Network
		Bayesian Networks Are Belief Networks, Not Causal Networks
		Keep This in Mind About Bayesian Networks
		Chains, Forks, and Colliders
		Given a Data Set, How Do We Set Up a Bayesian Network for the Involved Variables?
	Graph Diagrams for Probabilistic Causal Modeling
	A Brief History of Graph Theory
	Main Considerations in Graph Theory
		Spanning Trees and Shortest Spanning Trees
		Cut Sets and Cut Vertices
		Planarity
		Graphs as Vector Spaces
		Realizability
		Coloring and Matching
		Enumeration
	Algorithms and Computational Aspects of Graphs
	Summary and Looking Ahead
Chapter 10. Operations Research
	No Free Lunch
	Complexity Analysis and O() Notation
	Optimization: The Heart of Operations Research
	Thinking About Optimization
		Optimization: Finite Dimensions, Unconstrained
		Optimization: Finite Dimensions, Constrained Lagrange Multipliers
		Optimization: Infinite Dimensions, Calculus of Variations
	Optimization on Networks
		Traveling Salesman Problem
		Minimum Spanning Tree
		Shortest Path
		Max-Flow Min-Cut
		Max-Flow Min-Cost
		The Critical Path Method for Project Design
	The n-Queens Problem
	Linear Optimization
		The General Form and the Standard Form
		Visualizing a Linear Optimization Problem in Two Dimensions
		Convex to Linear
		The Geometry of Linear Optimization
		The Simplex Method
		Transportation and Assignment Problems
		Duality, Lagrange Relaxation, Shadow Prices, Max-Min, Min-Max, and All That
		Sensitivity
	Game Theory and Multiagents
	Queuing
	Inventory
	Machine Learning for Operations Research
	Hamilton-Jacobi-Bellman Equation
	Operations Research for AI
	Summary and Looking Ahead
Chapter 11. Probability
	Where Did Probability Appear in This Book?
	What More Do We Need to Know That Is Essential for AI?
	Causal Modeling and the Do Calculus
		An Alternative: The Do Calculus
	Paradoxes and Diagram Interpretations
		Monty Hall Problem
		Berkson’s Paradox
		Simpson’s Paradox
	Large Random Matrices
		Examples of Random Vectors and Random Matrices
		Main Considerations in Random Matrix Theory
		Random Matrix Ensembles
		Eigenvalue Density of the Sum of Two Large Random Matrices
		Essential Math for Large Random Matrices
	Stochastic Processes
		Bernoulli Process
		Poisson Process
		Random Walk
		Wiener Process or Brownian Motion
		Martingale
		Levy Process
		Branching Process
		Markov Chain
		Itô’s Lemma
	Markov Decision Processes and Reinforcement Learning
		Examples of Reinforcement Learning
		Reinforcement Learning as a Markov Decision Process
		Reinforcement Learning in the Context of Optimal Control and Nonlinear Dynamics
		Python Library for Reinforcement Learning
	Theoretical and Rigorous Grounds
		Which Events Have a Probability?
		Can We Talk About a Wider Range of Random Variables?
		A Probability Triple (Sample Space, Sigma Algebra, Probability Measure)
		Where Is the Difficulty?
		Random Variable, Expectation, and Integration
		Distribution of a Random Variable and the Change of Variable Theorem
		Next Steps in Rigorous Probability Theory
		The Universality Theorem for Neural Networks
	Summary and Looking Ahead
Chapter 12. Mathematical Logic
	Various Logic Frameworks
	Propositional Logic
		From Few Axioms to a Whole Theory
		Codifying Logic Within an Agent
		How Do Deterministic and Probabilistic Machine Learning Fit In?
	First-Order Logic
		Relationships Between For All and There Exist
	Probabilistic Logic
	Fuzzy Logic
	Temporal Logic
	Comparison with Human Natural Language
	Machines and Complex Mathematical Reasoning
	Summary and Looking Ahead
Chapter 13. Artificial Intelligence and Partial Differential Equations
	What Is a Partial Differential Equation?
	Modeling with Differential Equations
		Models at Different Scales
		The Parameters of a PDE
		Changing One Thing in a PDE Can Be a Big Deal
		Can AI Step In?
	Numerical Solutions Are Very Valuable
		Continuous Functions Versus Discrete Functions
		PDE Themes from My Ph.D. Thesis
		Discretization and the Curse of Dimensionality
		Finite Differences
		Finite Elements
		Variational or Energy Methods
		Monte Carlo Methods
	Some Statistical Mechanics: The Wonderful Master Equation
	Solutions as Expectations of Underlying Random Processes
	Transforming the PDE
		Fourier Transform
		Laplace Transform
	Solution Operators
		Example Using the Heat Equation
		Example Using the Poisson Equation
		Fixed Point Iteration
	AI for PDEs
		Deep Learning to Learn Physical Parameter Values
		Deep Learning to Learn Meshes
		Deep Learning to Approximate Solution Operators of PDEs
		Numerical Solutions of High-Dimensional Differential Equations
		Simulating Natural Phenomena Directly from Data
	Hamilton-Jacobi-Bellman PDE for Dynamic Programming
	PDEs for AI?
	Other Considerations in Partial Differential Equations
	Summary and Looking Ahead
Chapter 14. Artificial Intelligence, Ethics, Mathematics, Law, and Policy
	Good AI
	Policy Matters
	What Could Go Wrong?
		From Math to Weapons
		Chemical Warfare Agents
		AI and Politics
		Unintended Outcomes of Generative Models
	How to Fix It?
		Addressing Underrepresentation in Training Data
		Addressing Bias in Word Vectors
		Addressing Privacy
		Addressing Fairness
		Injecting Morality into AI
		Democratization and Accessibility of AI to Nonexperts
		Prioritizing High Quality Data
	Distinguishing Bias from Discrimination
	The Hype
	Final Thoughts
Index
About the Author
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