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Mathematical foundations

The notation an AI system is written in, taught as a modelling language rather than a prerequisite to survive.

  1. Mathematics as a Modeling Language
  2. Sets, Functions, Relations, and Notation
  3. Scalars, Units, and Coordinate Systems
  4. Mathematical Arguments, Assumptions, and Counterexamples
  5. Vectors, Inner Products, and Norms
  6. Orthogonality, Projections, and Least Squares
  7. Matrices and Linear Transformations
  8. Linear Systems, Rank, and Null Spaces
  9. Determinants, Volume, and Change of Coordinates
  10. Quadratic Forms and Positive Definiteness
  11. Eigenvalues, Eigenvectors, and Spectral Thinking
  12. Singular Value Decomposition and Low-Rank Structure
  13. Tensors, Shapes, and Broadcasting
  14. Kernels, Gram Matrices, and Implicit Feature Spaces
  15. Graphs, Laplacians, and Spectral Structure
  16. Derivatives and Local Linearization
  17. Gradients, Jacobians, and Hessians
  18. Taylor Expansions, Remainders, and Curvature
  19. The Chain Rule, Computational Graphs, and Automatic Differentiation
  20. Matrix Calculus, JVPs, VJPs, and Hessian Products
  21. Objectives, Stationary Points, and Local Optima
  22. Convexity, Smoothness, and Optimization Landscapes
  23. Gradient Methods, Conditioning, and Preconditioning
  24. Stochastic Gradients and Noisy Optimization
  25. Constraints, Lagrange Multipliers, and Duality
  26. Proximal Methods, Nonsmooth Objectives, and Sparsity
  27. Random Variables and Probability Models
  28. Joint, Marginal, and Conditional Probability
  29. Expectation, Variance, Covariance, and Gaussian Geometry
  30. Common Distributions and Generative Assumptions
  31. Transformations of Random Variables and Change of Variables
  32. Laws of Large Numbers, Central Limit Theorems, and Concentration
  33. Monte Carlo, Importance Sampling, and MCMC
  34. Markov Chains, Stationarity, and Stochastic Processes
  35. Point Estimation: Bias, Variance, Consistency, and Efficiency
  36. Bayes’ Rule, Priors, Likelihoods, and Posteriors
  37. Maximum Likelihood, MAP, and Posterior Prediction
  38. Confidence Intervals, Hypothesis Tests, and Bootstrap
  39. Entropy, Cross-Entropy, KL Divergence, and Mutual Information
  40. Proper Scoring Rules, Calibration, and Decision Theory
  41. Empirical Risk, Selection, and Generalization
  42. Capacity, VC Dimension, Rademacher Complexity, and Stability
  43. Numerical Stability and Finite Precision
  44. Mathematical Foundations Capstone: Audit a Learning System