Mathematics for AI/ML/DL - Overview

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Mathematics for AI/ML/DL - Overview

Introduction

A solid mathematical foundation is essential for deeply understanding and effectively utilizing artificial intelligence, machine learning, and deep learning. This course systematically presents the core mathematical concepts required for AI/ML/DL.

This course covers mathematical fields that form the theoretical foundation of AI, including linear algebra, calculus, probability theory, optimization theory, and information theory. Each lesson is designed with theoretical explanations alongside Python code examples, allowing you to implement and visualize mathematical concepts in practice.

The goal is not simply to memorize formulas, but to understand why this mathematics is necessary and how it applies to ML algorithms.

File List

No. Filename Topic Main Content
00 00_Overview.md Overview Course introduction and learning guide
01 01_Vectors_and_Matrices.md Vectors and Matrices Vector spaces, basis, matrix operations, linear transformations
02 02_Matrix_Decompositions.md Matrix Decompositions Eigendecomposition, SVD, PCA, LU/QR decomposition
03 03_Matrix_Calculus.md Matrix Calculus Jacobian, Hessian, backpropagation mathematics
04 04_Norms_and_Distances.md Norms and Distances Lp norms, cosine similarity, distance metrics
05 05_Multivariate_Calculus.md Multivariate Calculus Partial derivatives, gradients, directional derivatives, Taylor series
06 06_Optimization_Fundamentals.md Optimization Fundamentals Convex functions, Lagrange multipliers, KKT conditions
07 07_Gradient_Descent_Theory.md Gradient Descent Theory GD convergence analysis, SGD, momentum, Adam
08 08_Probability_for_ML.md Probability for ML Random variables, expectation, variance, Bayes' theorem
09 09_Maximum_Likelihood_and_MAP.md MLE and MAP MLE, MAP, relationship with regularization
10 10_Information_Theory.md Information Theory Entropy, cross-entropy, KL divergence, mutual information
11 11_Probability_Distributions_Advanced.md Advanced Distributions Exponential family, multivariate Gaussian, conjugate priors
12 12_Sampling_and_Monte_Carlo.md Sampling and Monte Carlo MCMC, Gibbs sampling, reparameterization trick
13 13_Linear_Algebra_for_Deep_Learning.md Linear Algebra for DL Tensors, einsum, broadcasting, numerical stability
14 14_Convexity_and_Duality.md Convexity and Duality Convex optimization, Lagrange duality, proximal operators
15 15_Graph_Theory_and_Spectral_Methods.md Graph Theory and Spectral Graph Laplacian, spectral clustering, GNN mathematics
16 16_Manifold_and_Representation_Learning.md Manifold Learning Manifold hypothesis, geodesics, t-SNE/UMAP mathematics
17 17_Math_of_Attention_and_Transformers.md Mathematics of Attention Self-attention, positional encoding, multi-head attention
18 18_Math_of_Generative_Models.md Mathematics of Generative Models VAE ELBO, GAN objective, diffusion model mathematics

Required Libraries

To run the code examples in this course, the following libraries are required:

pip install numpy scipy matplotlib sympy torch
  • NumPy: Vector and matrix operations, linear algebra
  • SciPy: Optimization, probability distributions, special functions
  • Matplotlib: Visualization of mathematical concepts
  • SymPy: Symbolic calculus, formula expansion
  • PyTorch: Automatic differentiation, deep learning math implementation

Phase 1: Linear Algebra Fundamentals (01-05) - 2-3 weeks

  • Basic concepts of vectors and matrices
  • Matrix decompositions and PCA
  • Matrix calculus
  • Norms and distance metrics
  • Multivariate calculus

Goal: Establish linear algebra fundamentals to understand the mathematical representation of deep learning models

Phase 2: Optimization Theory (06-07) - 1-2 weeks

  • Formulation of optimization problems
  • Convex optimization
  • Gradient descent and variants

Goal: Understand the working principles and convergence conditions of learning algorithms

Phase 3: Probability Theory and Information Theory (08-12) - 2-3 weeks

  • Probability fundamentals
  • Maximum likelihood estimation and MAP
  • Core concepts of information theory
  • Advanced probability distributions
  • Sampling techniques

Goal: Acquire probabilistic modeling and uncertainty quantification capabilities

Phase 4: Advanced Topics (13-18) - 2-3 weeks

  • Deep learning-specialized linear algebra
  • Convex duality
  • Graph neural network mathematics
  • Manifold learning
  • Transformer and generative model mathematics

Goal: Understand the theoretical foundations of modern AI models

Prerequisites

Required

  • High school mathematics: Calculus basics (limits, derivatives, integrals), matrix basics
  • Python programming: Basic syntax, functions, lists/dictionaries
  • Mathematical thinking: Logical reasoning, reading and interpreting formulas
  • NumPy basics: Array creation, indexing, basic operations
  • Calculus: Partial derivatives, chain rule
  • Linear algebra: Concepts of vectors, matrices, determinants

Prerequisite Courses

  • Python basics course
  • NumPy introduction

Learning Objectives

Upon completing this course, you will be able to:

  1. Master linear algebra: Understand vector spaces, matrix decompositions, linear transformations and apply them to ML problems
  2. Understand optimization theory: Grasp the mathematical principles and convergence conditions of gradient descent
  3. Think probabilistically: Mathematically model uncertainty and perform Bayesian inference
  4. Apply information theory: Design loss functions using entropy and KL divergence
  5. Implement backpropagation: Derive gradient computation formulas using matrix calculus
  6. Understand dimensionality reduction: Understand the mathematical principles and implementation of PCA and SVD
  7. Numerical stability: Recognize and resolve numerical issues that arise during computation
  8. Transformer mathematics: Understand the mathematical foundations of self-attention and positional encoding
  9. Generative model theory: Derive VAE ELBO and diffusion model objective functions
  10. Read papers: Independently understand formulas and proofs in AI papers

Course Features

Balance Between Theory and Practice

Each lesson provides mathematical proofs along with Python implementations. You can build intuition by not just looking at formulas but implementing and visualizing them in code.

ML/DL-Centric Approach

The focus is on how mathematics is actually used in machine learning and deep learning, not abstract mathematics. For example, when learning eigenvalue decomposition, we also cover applications like PCA and spectral clustering.

Modern Topics Included

Beyond traditional mathematics courses, we cover the mathematical foundations of cutting-edge AI models like Transformers, diffusion models, and graph neural networks.

Emphasis on Visualization

Rich visualizations using Matplotlib are provided to understand abstract concepts. Even high-dimensional space concepts can be intuitively understood through 2D/3D visualization.

Learning Strategies

1. Derive Formulas by Hand

Don't just read formulas in papers or textbooksโ€”write them down on paper and derive them step by step. Where you get stuck is exactly your learning point.

2. Validate with Code

After deriving a formula, always implement it in code to verify the results. You can confirm the meaning of the formula through numerical examples.

3. Build Intuition Through Visualization

Even high-dimensional data or complex functions can be visualized through appropriate cross-sections or projections. Understand the geometric meaning of mathematical concepts while creating graphs.

4. Practice Problems Are Essential

Don't skip the practice problems in each lesson. Even if you think you understand the concept, you can only verify true understanding by solving problems.

5. Practice Reading Papers

When you reach Phase 4, select an AI paper of interest and intensively analyze the formula sections. This is good practice for applying learned mathematics in real situations.

Learning Paths by Difficulty Level

Beginners (Weak math background)

  1. Study introductory linear algebra textbook (Gilbert Strang) in parallel
  2. Focus on lessons 01-02 (2-3 weeks)
  3. Study lessons 05, 08-09
  4. Refer to remaining lessons as needed

Intermediate (Math background available)

  1. Complete Phase 1-3 in normal order
  2. Selectively study Phase 4 based on areas of interest
  3. Complete entire course in 6-8 weeks

Advanced (Strong math background)

  1. Quick review of 01-05
  2. Focus on 06-07, 10, 14
  3. Advanced study of 13, 15-18
  4. Parallel paper formula derivation project

Project Ideas

Project suggestions to apply what you've learned:

  1. PCA-based face recognition: Eigenface implementation using SVD
  2. Gradient descent visualization tool: Compare various optimization algorithms
  3. Bayesian linear regression: Visualize prior/posterior distributions
  4. Information theory-based feature selection: Variable selection based on mutual information
  5. Transformer from scratch: Mathematical implementation of attention mechanism
  6. Simple diffusion model: Mathematical derivation and implementation of DDPM

References

Textbooks

  1. Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press. (especially Ch 2-4)
  2. Concise summary of essential deep learning mathematics
  3. Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.
  4. The bible of optimization theory
  5. Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer.
  6. Probabilistic perspective on machine learning
  7. Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
  8. Classic linear algebra introduction
  9. Murphy, K. P. (2022). Probabilistic Machine Learning: An Introduction. MIT Press.
  10. ML mathematics from a modern perspective

Online Courses

  1. 3Blue1Brown - Essence of Linear Algebra: The pinnacle of linear algebra visualization
  2. Gilbert Strang - MIT 18.06: Legendary linear algebra lectures
  3. Stanford CS229: Andrew Ng's machine learning math materials
  4. Fast.ai - Computational Linear Algebra: Practice-oriented approach

Papers and Blogs

  1. Distill.pub: ML math explained with interactive visualizations
  2. The Matrix Calculus You Need For Deep Learning (Parr & Howard, 2018)
  3. Understanding the difficulty of training deep feedforward neural networks (Glorot & Bengio, 2010)

Tools

  1. Wolfram Alpha: Formula calculation and verification
  2. Desmos: Function visualization
  3. GeoGebra: Geometric intuition development
  4. Jupyter Notebook: Interactive math notebooks

Version Information

  • First written: 2026-02-07
  • Author: Claude (Anthropic)
  • Python version: 3.8+
  • Major library versions:
  • NumPy >= 1.20
  • SciPy >= 1.7
  • Matplotlib >= 3.4
  • SymPy >= 1.9
  • PyTorch >= 1.10

License

This material is freely available for educational purposes. Please cite the source for commercial use.

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