Curriculum Map
How to Use This Series
Read the first 12 parts in order if you want a complete mathematical foundation. If you already know the basics, use the roadmap below to jump directly to the topic that unlocks the AI system you are trying to understand.
Phase 1
Foundations
Build fluency with mathematical notation, abstraction, set language, and counting arguments that appear everywhere in ML and AI.
Start Foundations →
Phase 2
Uncertainty
Understand distributions, inference, model evaluation, entropy, cross-entropy, KL divergence, and the loss functions behind modern learning.
Start Uncertainty →
Phase 3
Model Training
Connect vectors, matrices, optimization, backpropagation, loss functions, regularization, and numerical computing to trainable models.
Start Training Math →
Phase 4
Applications
Use advanced probability and from-scratch projects to make the theory executable: regression, Naive Bayes, PCA, and neural networks.
Start Applications →
12-Part Main Series
All Articles in This Series
The complete foundation: mathematical thinking, discrete math, probability, statistics, information theory, linear algebra, calculus, ML-specific math, computational Python, advanced topics, and capstone implementations.
Part 1
Mathematical Thinking
Abstraction, notation fluency, proof vs intuition, approximation, and the function families powering every ML model.
Read Part 1 →
Part 2
Set Theory & Foundations
Sets, subsets, power sets, De Morgan's Laws, Cartesian products, partitions, and similarity measures for ML.
Read Part 2 →
Part 3
Combinatorics
Counting principles, factorials, permutations, combinations, binomial theorem, and why AI search spaces explode.
Read Part 3 →
Part 4
Probability Fundamentals
Sample spaces, Bayes' theorem, random variables, key distributions, expectation, variance, and probabilistic models.
Read Part 4 →
Part 5
Statistics
Descriptive statistics, CLT, hypothesis testing, p-values, confidence intervals, correlation, and model evaluation.
Read Part 5 →
Part 6
Information Theory
Entropy, mutual information, cross-entropy, KL divergence, VAEs, GANs, and language-model evaluation ideas.
Read Part 6 →
Part 7
Linear Algebra
Vectors, matrices, linear transformations, eigendecomposition, SVD, PCA, distance metrics, and self-attention.
Read Part 7 →
Part 8
Calculus & Optimization
Derivatives, partial derivatives, Jacobians, Hessians, gradient descent, optimizers, backpropagation, and convexity.
Read Part 8 →
Part 9
ML-Specific Math
Loss functions, regularization, bias-variance tradeoff, MLE, MAP, softmax gradients, activations, and batch norm.
Read Part 9 →
Part 10
Computational Math
NumPy vectorization, SciPy optimization and stats, SymPy symbolic math, Monte Carlo methods, and integration.
Read Part 10 →
Part 11
Advanced Topics
Multivariate Gaussian distributions, Bayesian inference, Markov chains, EM algorithm, concentration inequalities, and PAC learning.
Read Part 11 →
Part 12
Projects & Applications
Build linear regression, Naive Bayes, PCA, and a two-layer neural network from scratch using only NumPy.
Read Part 12 →
7-Part Modern AI Extension
From Foundations to GenAI Systems
These extensions turn the existing mathematical foundation into direct fluency with transformers, LLMs, retrieval, diffusion, alignment, and modern training systems.
Extension 1
Tensor Calculus & Autodiff
Tensors, broadcasting semantics, computational graphs, forward-mode and reverse-mode autodiff, vector-Jacobian products, and gradient flow.
Read Extension →
Extension 2
Transformer & LLM Math
Tokenization, embeddings, positional encodings, scaled dot-product attention, multi-head attention, residuals, layer norm, masking, and KV cache complexity.
Read Extension →
Extension 3
Generative Model Math
ELBO derivations, latent spaces, GAN objectives, normalizing flows, score matching, denoising objectives, and classifier-free guidance.
Read Extension →
Extension 4
Training, Alignment & Evaluation
AdamW, schedulers, gradient clipping, calibration, perplexity, preference optimization, RLHF, DPO, uncertainty, and benchmark confidence.
Read Extension →
Extension 5
Reinforcement Learning Mathematics
MDPs, Bellman equations, policy gradient theorem, advantage estimation (GAE), and PPO clipping — the full RL math stack from first principles.
Read Extension →
Extension 6
Contrastive & Metric Learning
Triplet loss, InfoNCE, NT-Xent, and CLIP objective — derive the mathematics of self-supervised representation learning from information theory.
Read Extension →
Extension 7
Explainability & Attribution Math
Shapley values from game theory, LIME local approximation, integrated gradients via path integrals, and influence functions — the complete XAI math toolkit.
Read Extension →
4 Capstone Upgrades
Best-in-Class Project Path
After the NumPy capstones in Part 12, these projects are the next practical bridge from math foundations to modern AI engineering.
Project
Self-Attention from Scratch
Implement scaled dot-product attention with NumPy and verify tensor shapes, causal masks, and square-root scaling.
Read Capstone →Project
Tiny Next-Token Model
Connect probability, cross-entropy, softmax temperature, and autoregressive generation in a toy predictor.
Read Capstone →Project
Embedding Search Mini-RAG
Use vector similarity, normalization, nearest-neighbor search, and ranking metrics to build the mathematical core of RAG.
Read Capstone →Project
LoRA as Low-Rank Adaptation
Connect SVD and rank-constrained updates to parameter-efficient fine-tuning and low-rank adapters.
Read Capstone →
4 Specialist Notes
Niche Math — Canonical Derivations
Short, focused derivation articles for domain-specific mathematics used by one or two consumer series. Each is 10–15 minutes and provides the canonical proof/derivation that consumer series link back to.
SN 1
NLP Smoothing Mathematics
Derive Laplace, absolute discounting, and Kneser-Ney smoothing — the probability theory that makes language models work on unseen n-grams.
Read Note →SN 2
Detection & Localization Geometry
IoU, GIoU, DIoU, CIoU losses and NMS — the geometric math powering object detection from set theory and optimization.
Read Note →SN 3
Scaling Laws & Compound Optimization
Power-law scaling, Chinchilla-optimal training, and EfficientNet compound scaling — the math guiding model size decisions.
Read Note →SN 4
Recommendation & Ranking Math
Matrix factorization, ALS convergence, BPR pairwise loss, and LambdaRank — recommendation systems from optimization theory.
Read Note →