Mathematical Foundations for AI/ML

Machine learning is built on mathematics.
If you want to truly understand AI/ML models — not just use them as black boxes — you need a strong grasp of the math that powers them.

This section is a self-contained math course for AI/ML, starting from the basics and going into the depth needed for modern ML research and practice.
Each lesson combines:

• Theory explained from first principles.
• ML connections showing why it matters.
• Python & Rust code for hands-on practice.

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📐 Linear Algebra (Data Representation & Transformation)

Linear algebra is the language of data. Vectors, matrices, and transformations form the backbone of ML.

1. Scalars, Vectors, and Matrices
2. Vector Operations: Dot Product, Norms, and Distances
3. Matrix Operations: Multiplication, Transpose, and Inverse
4. Special Matrices: Identity, Diagonal, Orthogonal
5. Rank, Determinant, and Inverses
6. Eigenvalues and Eigenvectors
7. Singular Value Decomposition (SVD)
8. Positive Semi-Definite Matrices and Covariance
9. Linear Transformations and Geometry
10. Subspaces and Basis
11. Linear Independence and Orthogonality
12. Projections and Least Squares
13. Matrix Factorizations in ML (LU, QR, Cholesky)
14. Pseudo-Inverse & Ill-Conditioned Systems
15. Block Matrices and Kronecker Products
16. Spectral Decomposition & Applications

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🔢 Calculus (Optimization & Learning)

Calculus is the mathematics of change, powering optimization, learning algorithms, and probability.

1. Functions and Limits: The Language of Change
2. Derivatives: Measuring Change
3. Partial Derivatives & Gradients
4. Chain Rule & Backpropagation
5. Higher-Order Derivatives - Hessian & Curvature
6. Convexity and Optimization Landscapes
7. Gradient Descent & Variants
8. Advanced Optimization (Momentum, Adam, RMSProp)
9. Constrained Optimization (Lagrange Multipliers, KKT)
10. Integration Basics - Area Under the Curve
11. Probability Meets Calculus - Continuous Distributions
12. Differential Equations in ML (Neural ODEs, Dynamics)
13. Taylor Series & Function Approximations
14. Multivariable Taylor Expansion & Quadratic Approximations
15. Integral Transforms (Laplace, Fourier) in ML
16. Measure Theory Lite - Probability on Solid Ground

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🎲 Probability (Uncertainty in ML)

Probability is the language of uncertainty. It allows ML models to quantify confidence, handle noise, and make predictions.

1. Why Probability in ML?
2. Random Variables & Distributions
3. Expectation, Variance & Covariance
4. Conditional Probability & Bayes Theorem
5. Independence & Correlation
6. Law of Large Numbers & Central Limit Theorem
7. Maximum Likelihood Estimation (MLE)
8. Maximum A Posteriori (MAP) Estimation
9. Entropy, Cross-Entropy & KL Divergence
10. Markov Chains & Sequential Models
11. Bayesian Inference for ML

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📊 Statistics (Inference & Model Evaluation)

Statistics is about learning from data — estimating, testing, and validating models.

1. Data Summaries: Mean, Median, Mode, Variance
2. Distributions in Practice - Normal, Binomial, Poisson
3. Correlation & Covariance in Data
4. Sampling & Sampling Distributions
5. Estimation & Confidence Intervals
6. Hypothesis Testing - p-values, t-tests, Chi-square
7. ANOVA & Comparing Multiple Groups
8. Resampling Methods - Bootstrap & Permutation Tests
9. Maximum Likelihood vs. Method of Moments
10. Bayesian Statistics in Practice
11. Bias, Variance & Error Decomposition
12. Cross-Validation & Resampling for ML
13. Statistical Significance in ML Experiments
14. Nonparametric Statistics - Beyond Distributions
15. Multivariate Statistics - Correlated Features & MANOVA
16. Time Series Basics - Trends, Seasonality, ARIMA
17. Causal Inference - Correlation vs. Causation
18. Experimental Design & A/B Testing in ML

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🔑 Miscellaneous Math (Advanced & Cross-Cutting Topics)

Some powerful mathematical ideas lie outside the standard silos but are crucial for deep ML/AI understanding.

1. Numerical Linear Algebra (Large-Scale Computations)

• Iterative Solvers - Conjugate Gradient, Power Method, Lanczos
• Numerical Stability & Conditioning in Linear Algebra
• Sparse Matrices & Efficient Computation

2. Concentration Inequalities (Generalization & Learning Theory)

• Markov, Chebyshev, Hoeffding Inequalities
• Jensens Inequality & Convex Functions
• Generalization Bounds in ML

3. Information Geometry & Natural Gradients

• Fisher Information Matrix
• Natural Gradient Descent
• Information Geometry in Variational Inference

4. Stochastic Processes & Random Dynamics

• Martingales & Random Walks
• Brownian Motion & Stochastic Differential Equations
• Applications in Reinforcement Learning & Diffusion Models

5. High-Dimensional Data & Geometry

• Curse of Dimensionality
• Johnson–Lindenstrauss Lemma & Random Projections
• High-Dimensional Statistics in ML

6. Extended Deep Dives

• Tensors & Tensor Operations
• Manifold Learning & Nonlinear Dimensionality Reduction
• Spectral Methods in ML (Graph Laplacians, Spectral Clustering)

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✅ Summary

By completing the Mathematical Foundations, you'll have the tools to:

• Represent and manipulate data (Linear Algebra)
• Understand optimization and learning (Calculus)
• Quantify uncertainty (Probability)
• Draw conclusions from data (Statistics)
• Handle advanced ML-specific math (Miscellaneous)

This math backbone makes the Core ML and Deep Learning modules much easier to understand — and gives you the confidence to tackle AI research.

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