Math and Architectures of Deep Learning

12 Manifolds, homeomorphism, and neural networks

This chapter covers

• Introduction to manifolds
• Introduction to homeomorphism
• Role of manifolds and homeomorphism in neural networks

This is a short chapter that briefly introduces (barely scratching the surface of) a topic that could fill an entire mathematics textbook. A rigorous or even comprehensive treatment of manifolds is beyond the scope of this book. Instead, this chapter primarily focuses on geometric intuitions required for deep learning.

12.1 Manifolds

A manifold is a generalization of the notions of curve, surface, and volume into a unified concept that works in arbitrary dimensions. In machine learning, the input space can be viewed as a manifold. Usually, the input manifold is not ...