Basic Topology and Geometry for Differentiable Manifolds

Moving from Euclidean Space to Non-Euclidean Space

Ecludiean space is very intuitive. Most of objects in the real world are in Euclidean measure which can be understood naturally. However, when we begin to talk about machine learning models that usually have millions or trillions of parameters, it becomes impossible to deciper them in traditional way. To study latent variables or lower dimesional spaces learned from high dimenisonal input data, we need to equip ourselves with more abstract tools to analyze them. In addition, this would help us understand what model actually learn from data and make it possible to add strong inductive biases from existing knowledge.

Metric Space

A metric space is a set equipped with a function that satisfies the following properties:

• Non-negativity: $d(x,y)\ge 0$ for all $x,y$ in the set
• Identity of indiscernibles: $d(x,y)=0$ if and only if $x=y$
• Symmetry: $d(x,y)=d(y,x)$
• Triangle inequality: $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z$ in the set