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Mathematical Background

This guide covers the essential mathematical concepts needed to understand how deep learning frameworks work, with a focus on derivatives and the chain rule.

Basic Calculus: Understanding Derivatives​

What is a Derivative?​

  • A derivative measures how much a function's output changes when we make a tiny change to its input
  • In deep learning, we use derivatives to understand how changes in the network's parameters affect its predictions

The Chain Rule: A Lens Analogy​

Imagine light passing through a series of lenses:

  • First lens magnifies the image 2x
  • Second lens reduces the image to 1/3 of its size
  • Final lens magnifies the image 4x

The final image size is: original size × 2 × (1/3) × 4 = 8/3 times the original

This is exactly how the chain rule works in calculus:

  • Each function (like each lens) transforms its input
  • The total effect is the product of all individual transformations
  • In deep learning, we use this to calculate how each layer's parameters affect the final output

Essential Linear Algebra​

For implementing a deep learning framework, you mainly need to understand:

  • Matrix multiplication for layer operations
  • Transpose operations for backpropagation
  • Basic vector operations

Quick Note on Probability​

While probability theory is crucial for understanding machine learning concepts, for building a basic deep learning framework, you mainly need to know:

  • How to normalize values (making them sum to 1)
  • Basic concepts of random initialization

Resources​

Mathematics for Machine Learning: Linear Algebra on Coursera by Imperial College London is a good resource for refreshing your linear algebra knowledge.