Core Mathematical Foundations of Deep Learning#

The central problem of deep learning—enabling machines to learn patterns from data—has a mathematical answer. This chapter reveals the mechanisms behind it.

Chapter overview

Section

Content

Key takeaway

Introduction

Problem introduction and overview

Why do we need these mathematical tools?

Computational graph

Computational graph principles

Visual representation of complex computations

Learning Task Formulations

Task types and optimization objectives

Mathematical formulations of classification vs. regression

Activation Functions

Activation functions in detail

Introducing non-linearity and decision boundaries

Loss Functions

Loss function design

How to quantify prediction errors

Backpropagation Algorithm

Backpropagation algorithm

The chain rule for efficient gradient computation

Gradient Descent and Optimization Algorithms

Gradient descent optimization

Iterative strategies for parameter updates

Summary and Outlook

Summary and outlook

Transitioning from theory to practice

Learning path

This chapter serves as the theoretical foundation for the entire series:

        graph LR
    A[建立直觉<br/>几何直观] --> B[理解原理<br/>数学机制]
    B --> C[动手实践<br/>代码实现]
    C --> D[后续章节<br/>架构与应用]

Core insight: We do not dive into complex network architectures here, but focus instead on the core mathematical mechanisms that make neural networks “work”—with code implementations corresponding to each concept.

Prerequisites

This chapter is the theoretical foundation of the deep learning series. It requires no prior deep learning knowledge, but assumes:

  • Basic Python programming: Familiarity with fundamental syntax and functions

  • High-school mathematics: Concepts of functions and derivatives

  • NumPy basics: Basic understanding of array operations is recommended

Contributors and Revision History

View detailed revision history
  • dc35df2 2026-06-14 - Heyan Zhu: refactor: unify into two-chapter mountain-climbing narrative
  • f1c39fd 2026-05-11 - Heyan Zhu: docs: deduplicate index & introduction; fix the latex export problem on index content structures
  • 2dc78ed 2026-05-09 - Heyan Zhu: fix: address latex errors when building pdf
  • f159d56 2026-05-08 - Heyan Zhu: fix: address complaints from the linter
  • bba351e 2026-04-29 - Heyan Zhu: docs: update chapter summaries and learning paths for consistency
  • 0cdb1e4 2026-04-29 - Heyan Zhu: feat: add model-serving chapter and update related content
  • 59126f4 2026-04-26 - Heyan Zhu: docs(math-fundamentals): update content structure and add citations
  • ae2053f 2026-04-26 - Heyan Zhu: docs(math-fundamentals): add task-formulations and update related content
  • 756a793 2026-04-25 - Heyan Zhu: docs(math-fundamentals): update content structure and improve explanations
  • 0c291d7 2025-12-10 - Heyan Zhu: docs: restructure course materials and add new content