12 Manifolds, homeomorphism and Neural Networks

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

12.1 Manifolds

Manifolds are characterized by three properties, each of which will be explained below. They are locally Euclidean, Haussdorf and second countable.

12.1.1 Locally Euclidean

12.1.2 Hausdorff

12.1.3 Second Countable

12.2 Homeomorphism

12.3 Neural Networks and homeomorphism between manifolds

12.4 Chapter Summary