Essential Math for AI Hala Nelson

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• 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
  • OReilly Online Learning
  • How to Contact Us
  • Acknowledgments
• 1. Why Learn the Mathematics of AI?
  • What Is AI?
  • Why Is AI So Popular Now?
  • What Is AI Able to Do?
    • An AI Agents Specific Tasks
  • What Are AIs 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
• 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
• 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
      • The predicted value versus the true value
      • The absolute value distance versus the squared distance
      • Functions with singularities
      • For linear regression, the loss function is the mean squared error
      • Notation: Vectors in this book are always column vectors
      • The training, validation, and test subsets
      • Recap
      • When the training data has highly correlated features
    • Optimization
      • Convex landscapes versus nonconvex landscapes
      • How do we locate minimizers of functions?
      • Calculus in a nutshell
      • A one-dimensional optimization example
      • Derivatives of linear algebra expressions that we use all the time
      • Minimizing the mean squared error loss function
  • 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
      • Training function
      • Loss function
      • Optimization
      • The kernel trick
    • Decision Trees
      • Entropy and Gini impurity
      • Entropy and information gain
        • Binary output
        • Multi-class output
      • Gini impurity
      • Regression decision trees
      • Shortcomings of decision trees
    • Random Forests
    • k-means Clustering
  • Performance Measures for Classification Models
  • Summary and Looking Ahead
• 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
      • The weights
      • A linear combination plus bias
      • Pass the result through a nonlinear activation function
      • Notation overview
    • Common Activation Functions
    • Universal Function Approximation
      • Example 1: Approximating irrational numbers with rational numbers
      • Example 2: Approximating continuous functions with polynomials
      • Statement of the universal approximation theorem for neural networks
    • Approximation Theory for Deep Learning
  • Loss Functions
  • Optimization
    • Mathematics and the Mysterious Success of Neural Networks
    • Gradient Descent i+1 = i - L ( i )
    • Explaining the Role of the Learning Rate Hyperparameter
      • The scale of the features affects the performance of the gradient descent
      • Near the minima (local and/or global), flat regions, or saddle points of the loss functions landscape, the gradient descent method crawls
    • Convex Versus Nonconvex Landscapes
    • Stochastic Gradient Descent
    • Initializing the Weights 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
      • Commonly used weight decay regularizations
      • When do we use plain linear regression, ridge, lasso, or elastic net?
    • Penalizing the l 2 Norm Versus Penalizing the l 1 Norm
    • Explaining the Role of the Regularization Hyperparameter
  • Hyperparameter Examples That Appear in Machine Learning
  • Chain Rule and Backpropagation: Calculating L ( 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
• 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
• 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
      • Rotation matrix
      • Reflection matrix
    • 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
• 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
  • Zipfs 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 selection and dimension reduction
      • Shortcomings of 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
      • Word2vec vector representation of individual terms by incorporating continuous-ness attributes
      • How to visualize vectors representing words
      • Facebooks fastText vector representation of individual n-character grams
      • Doc2vec or par2vec vector representation of a document
      • Global vector or vector representation of words
  • 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
• 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)
      • Conditional independence
      • Universal approximation
    • The Original Autoencoder
  • Probabilistic Language Modeling
  • Summary and Looking Ahead
• 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
• 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
      • The meaning of Lagrange multipliers
    • Optimization: Infinite Dimensions, Calculus of Variations
      • Analogy between optimizing functions and optimizing functionals
      • Example 1: Harmonic functions, the Dirichlet energy, and the heat equation
        • A harmonic function minimizes the Dirichlet energy
        • The heat equation does gradient descent for the Dirichlet energy functional
      • Example 2: The shortest path between two points is along the straight line connecting them
      • Other introductory examples to the 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 interplay of algebra and geometry
    • The Simplex Method
      • The main idea of the simplex method
      • The simplex method hops around the corners of the polyhedron
      • Steps of the simplex method
      • Notes on the simplex method
      • The revised simplex method
      • The full tableau implementation of the simplex method
    • Transportation and Assignment Problems
    • Duality, Lagrange Relaxation, Shadow Prices, Max-Min, Min-Max, and All That
      • Motivation for duality-Lagrange multipliers
      • Finding the dual linear optimization problem from the primal linear optimization problem
      • Derivation for the dual of a linear optimization problem in standard form
      • Dual simplex method
      • Example: Networks, linear optimization, and duality
      • Example: Two-person zero-sum games, linear optimization, and duality
      • Quadratic optimization with linear constraints, Lagrangian, min-max theorem, and duality
    • Sensitivity
  • Game Theory and Multiagents
  • Queuing
  • Inventory
  • Machine Learning for Operations Research
  • Hamilton-Jacobi-Bellman Equation
  • Operations Research for AI
  • Summary and Looking Ahead
• 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
      • The adjustment formula
      • The backdoor criterion, or controlling for confounders
      • Controlling for confounders
      • Are there more rules that eliminate the do operator?
  • Paradoxes and Diagram Interpretations
    • Monty Hall Problem
    • Berksons Paradox
    • Simpsons Paradox
  • Large Random Matrices
    • Examples of Random Vectors and Random Matrices
      • Quantitative finance
      • Neuroscience
      • Mathematical physics: Wigner matrices
      • Multivariate statistics: Wishart matrices and covariance
      • Dynamical systems
      • Other equally important examples
    • 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
    • Itôs 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
      • Limit theorems
    • The Universality Theorem for Neural Networks
  • Summary and Looking Ahead
• 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
• 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
      • Discretize right away and do a computer simulation
      • The curse of dimensionality
      • The geometry of the problem
      • Model things that you care for
    • Discretization and the Curse of Dimensionality
    • Finite Differences
      • Example: Solve y ' ( x ) = 1 on [0,1], with boundary conditions y(0)=-1 and y(1)=0
      • Example: Discretize the one-dimensional heat equation u t = u xx in the interior of the interval x ( 0 , 1 )
    • 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
      • How does it work?
      • How do we use it to solve ODEs and PDEs?
      • Simple but very informative example
      • Where is the complication?
      • Recent successes!
      • Setting the stage for deep learning for PDEs
      • Mesh independence and different resolutions
  • AI for PDEs
    • Deep Learning to Learn Physical Parameter Values
    • Deep Learning to Learn Meshes
      • Deep learning for three-dimensional meshes
    • Deep Learning to Approximate Solution Operators of PDEs
      • Neural operator networks to learn the solution operators that we derived
      • The important questions
      • Fourier neural network
      • Statement of the universal approximation theorem for operators
      • How do we branch out and dive into the more technical details?
    • Numerical Solutions of High-Dimensional Differential Equations
    • Simulating Natural Phenomena Directly from Data
  • Hamilton-Jacobi-Bellman PDE for Dynamic Programming
    • Bellmans equation in deterministic and stochastic settings
    • The big picture
    • Hamilton-Jacobi-Bellman PDE
    • Solving the Hamilton-Jacobi-Bellman PDE
    • Dynamic programming and reinforcement learning
  • PDEs for AI?
  • Other Considerations in Partial Differential Equations
  • Summary and Looking Ahead
• 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

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