Mathematical Foundations of AI: A Complete Learning Path from Linear Algebra to Deep Learning

This article introduces the GitHub project Mathematics-of-AI, maintained by AaMRosas, covering seven major topics from singular value decomposition and machine learning basics to generative models and reinforcement learning. It provides theoretical explanations and code practices in Jupyter Notebook format, offering structured learning resources for learners who want to deeply understand the underlying principles of AI.

Mathematical foundations are key to entering deep learning in the AI field, but learners often feel overwhelmed by the complexity of mathematical concepts. The Mathematics-of-AI project aims to address this pain point by dividing topics in a progressive manner, helping to build a complete cognitive chain from mathematical principles to algorithm implementation.

The project is divided into seven modules:

1. Introduction to Deep Learning with Keras: Feedforward networks and MNIST tasks, combining Keras and JAX;
2. Singular Value Decomposition (SVD) and Introduction to NumPy: SVD principles and applications, basic NumPy operations;
3. Machine Learning Basics: SGD, bias-variance tradeoff, double descent phenomenon, SVM and kernel tricks;
4. Deep Network Architectures: Forward/backward propagation, CNN, RNN, Transformer;
5. Generative Models: VAE, GAN, diffusion models, sequence memory models;
6. Geometric Deep Learning: Symmetry groups, Graph Neural Networks (GNN);
7. Reinforcement Learning: MDP, Bellman equation, policy gradients, RLHF.

Technical Features:

• Jupyter Notebook interactive environment, learn while practicing;
• Combination of Keras and JAX, balancing concise API and performance;
• Equal emphasis on theory and practice, avoiding superficial learning;
• Progressive difficulty design.

Advantages over similar resources:

• Complete structure, covering from basics to cutting-edge;
• Runnable code, supporting local/Colab execution;
• Includes cutting-edge content such as RLHF and diffusion models.

For learners with different backgrounds:

• No prior knowledge: Complete all modules in order;
• Programming background but weak in math: Focus on modules 2 and 3;
• With ML experience: Skip module 1 and start from module 4;
• Researchers/engineers: Use as a reference manual to look up corresponding modules.

Limitations:

• Lack of prerequisite knowledge explanations (e.g., geometric deep learning requires group theory basics);
• No supporting exercises and solutions;
• Limited size of example datasets.

Improvement directions: Add prerequisite requirements, increase exercises, use larger datasets.

Mathematics-of-AI provides a clear roadmap for learners. A solid mathematical foundation is the key to distinguishing between superficial users and deep practitioners. It is suitable for students, transitioning engineers, and researchers to collect and study; understanding the underlying principles allows one to draw inferences and solve new problems.