فهرست مطالب

• Preface
• 1. Why Learn the Mathematics of AI?
• What Is AI?
• Why Is AI So Popular Now?
• What Is AI Able to Do?
• 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
• 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
• 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
• Logistic Regression: Classify into Two Classes
• Softmax Regression: Classify into Multiple Classes
• Incorporating These Models into the Last Layer of a Neural Network
• Other Popular Machine Learning Techniques and Ensembles of Techniques
• 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
• Loss Functions
• Optimization
• Regularization Techniques
• Hyperparameter Examples That Appear in Machine Learning
• Chain Rule and Backpropagation: Calculating ∇ L ( ω → i )
• Assessing the Significance of the Input Data Features
• Summary and Looking Ahead
• 5. Convolutional Neural Networks and Computer Vision
• Convolution and Cross-Correlation
• Convolution from a Systems Design Perspective
• Convolution and One-Dimensional Discrete Signals
• Convolution and Two-Dimensional Discrete Signals
• Linear Algebra Notation
• 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
• Three Ways to Multiply Matrices
• The Big Picture
• The Ingredients of the Singular Value Decomposition
• Singular Value Decomposition Versus the Eigenvalue Decomposition
• Computation of the Singular Value Decomposition
• 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
• Zipf’s Law for Term Counts
• Various Vector Representations for Natural Language Documents
• Cosine Similarity
• Natural Language Processing Applications
• Transformers and Attention Models
• Convolutional Neural Networks for Time Series Data
• Recurrent Neural Networks for Time Series Data
• 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
• 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
• Example: Machine Learning and Generative Networks for High Energy Physics
• Other Generative Models
• The Evolution of Generative Models
• 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
• Random Walks on Graphs
• Node Representation Learning
• Tasks for Graph Neural Networks
• Dynamic Graph Models
• Bayesian Networks
• Graph Diagrams for Probabilistic Causal Modeling
• A Brief History of Graph Theory
• Main Considerations in Graph Theory
• 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 on Networks
• The n-Queens Problem
• Linear Optimization
• 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
• Paradoxes and Diagram Interpretations
• Large Random Matrices
• Stochastic Processes
• Markov Decision Processes and Reinforcement Learning
• Theoretical and Rigorous Grounds
• Summary and Looking Ahead
• 12. Mathematical Logic
• Various Logic Frameworks
• Propositional Logic
• First-Order Logic
• 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
• Numerical Solutions Are Very Valuable
• Some Statistical Mechanics: The Wonderful Master Equation
• Solutions as Expectations of Underlying Random Processes
• Transforming the PDE
• Solution Operators
• AI for PDEs
• Hamilton-Jacobi-Bellman PDE for Dynamic Programming
• 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?
• How to Fix It?
• Distinguishing Bias from Discrimination
• The Hype
• Final Thoughts
• Index